GIFT  OF 


TABLES 


FOR 


ENGINEERING    CALCULATIONS 


BY 

RICHARD    C.    POWELL 

ASSOC.    MEM.  A.    I.   E.   E. 


SECOND      EDITION 
REVISED    AND     ENLARGED 


BERKELEY,     CAL. 
C.     POWELL,     PUBLISHER 
1907 


COPYRIGHT,    1907, 

BY     • 

RICHARD   C.   PCTWELL 


PRICE  $2,00 


PRESS  OF 

BERKELEY     REPORTER    INC. 

BOOKBINDERS    AND    PRINTERS 

BERKELEY,    CAL. 


PREFACE 

THE  present  voltfme  is  the  seco.nd  edition  of 
the    "Handbook   of   Tables    for   Engineering 
Calculations,"   compiled  by   Mr.   William   E. 
Hawley  and  the  author.    The  first  edition  has  been 
almost    entirely    rewritten    and    much    new    matter 
added,  being  nearly  doubled  in  size. 

Although  descriptions  and  illustrations  of  appa- 
ratus, and  discussions  of  best  practice  are  invalu- 
able to  the  engineer,  it  is  believed  that  there  is 
use  for  a  handbook  not  burdened  with  these  things, 
but  one  that  furnishes  to  the  student  and  engineer 
those  formulae  and  tables  which  enable  him  to 
perform  with  the  least  amount  of  time  and  effort 
the  calculations  incident  to  engineering. 

The  author  has  attempted  to  collect  the  most 
reliable  material  and  arrange  it  for  greatest  con- 
venience of  use;  those  tables  which  are  more  fre- 
quently in  simultaneous  use  have  been  put  together 
and  in  nearly  every  case  two-page  tables  are  on 
pages  which  face. 

Very  little  has  been  inserted  unrevised  with  the 
exception  of  the  tables  for  rolled  steel  shapes. 
These  the  author  was  enabled  to  reproduce  from 
the  "Cambria  Steel"  Handbook  through  the  gen- 
erosity of  the  Cambria  Steel  Company. 

The  author  has  had  the  advantage  of  many  sug- 
gestions, and  to  all  those  who  made  them,  and  also 
to  others  who  assisted  in  numerous  ways,  he  is 
very  grateful.  His  best  thanks  are  due  to  Pro- 
fessor LeConte  of  the  University  of  California  for 
his  assistance  in  the  preparation  of  the  section  on 
hydraulics,  which  was  for  the  most  part  compiled 
by  him. 

THE  AUTHOR. 
Berkeley,  August^  IJKfo  ^  o  * > 


CONTENTS 


Greek    Letters 2 

Logarithms 

Explanation    3 

Table— Natural   Logarithms   1.00    to    9.99 6 

Table— Natural  Logarithms  1  to  999 8 

Table — Common  Logarithms   1  to  9999 10 

Trigonometric   Functions 

Table — Natural  and  Logarithmic   Sine,   Cosine,   Tangent  and 

Cotangent     31 

Table — Minutes   in   Decimal   Parts   of   Degrees 40 

Table — Functions   of   Certain   Angles 49 

Table— Angles  Greater  than  90°  to  Angles  Less  than  90°...  40 

Formulae    41 

Any    Plane    Triangle    42 

Plane  Oblique  Triangles   43 

Right-Angled    Spherical    Triangles    44 

Any   Spherical   Triangle    44 

Circles 

Formulae — Arcs,  Cords  and  Segments  45 

1*able — Lengths  of  Circular  Arcs  for  Unit  Radius 46 

Table — Length  of  Cord,   Rise  and  Area  of  Segments 47 

Table — Circumferences  and  Areas  for  Diameters  in  Inches..  49 
Table — Areas  in  Square  Feet  for  Diameters  in  Inches 51 

Functions   of  the   Natural    Numbers 

Table — Square,    Cube,    Square    and    Cube    Roots,    Reciprocal, 

Area  and    Circumference   of   Circles    52 

Table— Fifth  Roots  and  Fifth  Powers    72 

Table — Square  and  Cube  Roots  of  Fractions   72 


CONTENTS  v 

Conversion   of   Units 

Table — Decimal  Equivalents  of  Fractions  of  One  Inch   73 

Table— Inches  in  Decimal  Parts  of  a  Foot 73 

Table— U.   S.  and  British  Units    74 

Table— Metric  Units    75 

Table — U.   S.  to  Metric  and  Metric  to  U.  S.  Lengths,  Areas, 

Volumes,  Weights,  Pressures  and  Miscellaneous   76 

Table — Conversion  Factors.  Angles,  Lengths,  Areas,  Vol- 
umes, Water  Discharges,  Weights,  Weights  per  Unit 
Length,  Densities,  Water  Weights  and  Volumes, 
Forces,  Work  and  Heat,  Power,  Angular  and  Linear 
Velocities,  Temperatures,  Electrostatic  to  Electromag- 
netic ' 78 

Wire   and   Sheet   Metal   Gauges    - 85 

Powers  and   Multiples  of  ir   e  and  g   86 

Hyperbolic    Functions 

Formulae    87 

Table— Tangent     87 

Table— Sine    88 

Table — Cosine  89 

Solution  of  Equations 

Quadratic   Equation— Algebraic  and   Slide   Rule    Solutions...  90 

Cubic   Equation — Slide   Rule   Solution    91 

Descartes'  Rule  of  Signs   92 

Homer's   Method    92 

Newton's    Method    94 

By  Plotting  or  by  the  Use  of  Tables  94 

Series    95 

Integrals 

Table— Integrals 96 

Definite   Integrals — Approximate   Evaluation    98 

Plane   Curves— Cycloid,    Logarithmic   Spiral   and   Catenary    99 


vi  CONTENTS 

Probable   Error 

Formulae    100 

Table— Constants    101 

Sections  and   Solids 

Guldin's  Rules  101 

Prismoidal  Formula  101 

Moments  of  Inertia  102 

Table — Area,  Center  of  Mass,  Moments  of  Inertia,  Radius  of 

Gyration  and  Section  Modulus  of  Sections  102 

Table — Surface,  Volume,  Center  of  Mass  and  Moments  of 

Inertia  of  Solids  107 

Coefficients  of  Strength  and  Deflection  for  Steel  .Shapes 109 

Properties  of  Simple  and  Compound  Rolled  Shapes 109 

Table— Properties  of  Standard  I-Beams  110 

Table — Properties  of  Bulb  Beams  112 

Table — Properties  of  Standard  Channels  114 

Table — Properties  of  Standard  Angles  116 

Table — Properties  of  T-bars  124 

Table— Properties  of  Z-bars  126 

Strength   of   Materials 

Table — Tension,  Compression,  Shear  and  Coefficient  of  Elas- 
ticity     128 

Columns — Formulae   128 

Beams 

Mechanics   of  Beams    129 

Table — Reactions,   Bending  Moment  and  Deflection   131 

Valency  and  Atomic  Weights  of  the  Elements— Table 134 

Densities    of    Substances — Table    134 

Heat 

Table — Comparison   of   Centigrade  and   Fahrenheit    136 

Table— Coefficients  of  Expansion  and   Specific  Heats 138 


CONTENTS  vii 

Heat— Con 't. 

Table— Density  and  Volume  of  Water 139 

Table— Melting   and   Boiling   Points    139 

Table — Boiling  Points  of  Water  for  Various  Pressures    139 

Table— Latent    Heats     140 

Gases — Formulae    140 

Table — Density  and  Constant  of  Gases   141 

Table— Specific  Heat  of  Gases    141 

Factor   of   Evaporation — Formulae    141 

Table — Factors    of   Evaporation    142 

Table — Properties  of  Saturated  Steam  ?r 143 

Table — Properties  of  Saturated  Vapor  of  Ammonia  150 

Table — Properties  of  Saturated  Vapor  of  Sulphur  Dioxide 150 

Magnetic   Constants 

Table — Magnetic  Properties  of  Iron  and  Steel   151 

Table — Loss  Due  to  Hysteresis    151 

Electrical   Constants 

Table — Specific  Resistances  and  Temperature  Coefficients 152 

Table— Specific    Inductive    Capacities    153 

Table— Rate  of  Electrolytic  Deposition    153 

Wires  and  Cables 

Table— Fusing  of  Wires    153 

Table— Properties  of  Copper  Wire   154 

Table — Properties  of  Aluminum  Wire   155 

Table — Properties  of  Copper  Cables   156 

Table — Properties  of  Aluminum  Cables  156 

Table — Diameters  of  Cotton-Covered  Wire   157 

B.  &  S.  Gauge — Law  and  Slide  Rule  Formulae 157 

Transmission    Line   Constants 

Formulae    158 

Table — Capacity,      Charging     Current     and     Reactance     for 

Ratios  of  Distance  between  Wires  to  Diameter    158 

Table — Capacity,  Charging  Current,  Reactance  and  Imped- 
ance of  Copper  and  Aluminum  for  Distances  between 
Wires.  Three  Phase  60  cycles 159 


viii  CONTENTS 

Transmission    Line   Constants — Con't. 

Table — Capacity,  Charging  Current,  Reactance  and  Imped- 
ance of  Copper  and  Aluminum  for  Distances  between 
Wires.  Single  Phase  25  cycles 160 

Hydraulics 

Immersed  Rectangle — Formulae    161 

Impact   Due  to  a  Jet — Formula , 161 

Orifices — Formulae     162 

Flow   Over  Weirs — Formulae    163 

Flow  Through  Pipes— Formulae   164 

Flow  in   Open   Channels — Formulae 165 

Table  I — Coefficients  of  Efflux  for  Circular  Vertical  Orifices.  .166 
Table  II — Coefficients  of  Efflux  for  Square  Vertical  Orifices.  .166 

Table  III — Imperfect    Contraction    167 

Table  IV— Coefficient  for  Conical  Orifices   167 

Table  V — Coefficients  for  Conical  Tubes 167 

Table  VI — Discharge  Coefficients  for  Weirs,  Francis'  For- 
mula   168 

Tables  VII,  VIII,  IX  and  X— Discharge  Coefficients  for  Weirs, 
Smith's  Formula  168 

3/ 
Table  XI— Values  of  h    /2. 169 

Table  XII— Weir  Measurement    170 

Table  XIII — Coefficients  of  Flow  for  Iron  Pipes 171 

Table  XIV— Loss  of  Head  by  Friction  in  Iron  Pipes 172 

Table  XV — Coefficients  of  Flow  for  Wood  Stave  Pipes 173 

Table  XVI— Coefficients  of  Friction  for  Elbows 174 

Table  XVII— Coefficients  of  Friction  for  Curves 174 

Table  XVIII— Kutter's   Coefficients    174 

Index     .  ..177 


GREEK  LETTERS. 


Greek  Letters 


A      a  Alpha 

B      /3  Beta 

7  Gamma 

5  Delta 

e  Epsilon 

f  Zeta 

T;  Eta 

0  Theta 

1  Iota 

K  Kappa 

X  Lambda 

M  Mu 


N 


0 


v  Nu 

£  Xi 

o  Omicron 

n     TT  Pi 

P       p  Rho 

2  <r   s  Sigma 

T      T  Tail 

T       v  Upsilon 

4>      0  Phi 

X     x  Chi 

*•     V  Psi 

1}      o>  Omega 


LOGARI1WMS 


LOGARITHMS 


Explanation 

The  logarithm  of  a  number  to  a  given  base  is  the  index  of 
the  power  to  which  the  base  must  be  raised  to  produce  the 
given  number.  If  a*  =  n,  x  is  the  logarithm  of  n  to  the  base 
a,  and  is  written  x  =  loga  n. 

The  only  systems  of  logarithms  in  practical  use  are  the  so- 
called  natural  (Hyperbolic)  and  the  common  (Briggs.)  In  the 

former  the  base  is  the  incommensurable  number  2.718282 

usually  denoted  by  e.  In  the  common  system  the  base  is  10. 
The  relations  between  logarithms  in  these  two  systems  are: 

Iog10  n  =  0.434294  log,  n,  log,  n  =  2.302585  log,0  n. 

The  natural  logarithm  is  often  designated  by  In;  the  com- 
mon by  log.  Thus,  In  x  is  read  natural  logarithm  of  x  while 
log  x  is  read  common  logarithm  of  x. 

The  logarithm  of  a  product  is  the  sum  of  the  logarithms 
of  its  factors,  thus 

log  (a  b  c    n)  =  log  a  -f  log  b  -f  log  c  + +  log  n. 

The  logarithm  of  a  quotient  is  the  logarithm  of  the  divi- 
dend minus  the  logarithm  of  the  divisor,  thus 

c 

log  =  log  a  —  log  b 


4  LOG  vRITHMS" 

The  logarithm  of  a  number  raised  to  any  power,  integral 
or  fractional,  is  the  logarithm.-  of  the  number  multiplied  by  the 
index  of  the  power,  thus 

log    nx  =••  x  log  n    and  log  * /  n  =   log  n 

These  rules  hold  for  any  base. 


Explanation  of  the  Tables. 

Natural  Logarithms.  The  natural  logarithms  of  the  num- 
bers from  1  to  9.99  are  given  on  pages  6  and  7,  and  from  1  to 
999  on  pages  8  and  9.  The  In  of  numbers  less  than  1  are  negative 


Common  Logarithms.  In  the  common  system,  the 
logarithm  is  considered  as  consisting  of  a  positive  decimal  part 
called  the  mantissa  and  an  integral  part  called  the  characteristic, 
which  may  be  either  positive  or  negative.  The  mantissa  is 
the  same  for  all  numbers  having  the  same  sequence  of  figures, 
and  hence  is  independent  of  the  decimal  point  or  of  ciphers 
which  preceed  or  follow  this  given  sequence  of  figures.  The 
characteristic  depends  upon  the  decimal  point  and  is  determined 
by  the  following  rules: 

The  characteristic  of  the  logarithm  of  a  number  greater  than  unity 
is  positive  and  is  less  by  one  than  the  number  of  digits  in  its  integral 
part. 

The  characteristic  of  the  logarithm  of  a  decimal  fraction  is  nega- 
tive and  is  greater  by  one  than  the  number  of  ciphers  immediately  after 
the  decimal  point . 

By  reversing  these  rules,  the  decimal  point  in  the  sequence 
of  figures  corresponding  to  any  mantissa  is  readily  found. 

The  logarithms  of  the  numbers  from  1  to  100,  and  the 
mantissas  of  the  numbers  from  100  to  10,000  are  given  on  pages 
10  to  30.  The  column  D  gives  the  tabular  differences,  and  where 


LOGARITHMS  5 

these  are  large,  the  proportional  parts  are  given  at  the  foot  of 
the  page. 

Examples.  The  mantissa  for  757.6  is  0.87944  and  the 
characteristic  is  2.  Hence,  log  757.6  =  2.87944. 

To  find  the  number  whose  log  is  1.70131,  we  first  find  the 
sequence  of  figures  corresponding  to  the  mantissa  0.70131, 
This  is  (page  21)  5027,  and  from  the  rule  for  the  characteristic 
we  find  the  number  to  be  50.27. 

The  mantissa  for  0.0438  is  0.64147  and  the  characteristic 
-2,  hence,  log  0.0438  ="2". 64147,  a  bar  being  placed  over  the 
characteristic  to  indicate  that  it  alone  is  negative.  To  avoid 
confusion,  it  is  better  to  add  and  subtract  10  from  the  log.  Thus 
the  log  0.0438  is  written 

log  0.0438  =8.64147  —  10. 

To  extract  the  root  of  a  decimal  fraction  we  add  and  subtract 
a  multiple  of  10;  as  for  example  to  find  the  cube  root  of  0.0438 
we  proceed  by  adding  and  subtracting  20  from  the  log  as  just 
written,  and  then  dividing  by  3,  thus 

3^28.641470  —  30 
9.547156—10 

The  figures  corresponding  to  the  mantissa  0.547156  (p.  17) 
are  easily  found  to  be  3525,  and  the  characteristic  determines 
the  number  to  be  0.3525. 

Trigonometric  Functions.  The  natural  sines,  tangents 
cotangents  and  cosines  with  their  logarithms  are  given  on  pages 
31  to  39.  The  values  are  for  angles  between  0  and  90  degrees, 
varying  by  tenths  of  degrees,  i.  e.,  by  6  minutes.  For  angles 
between  0  and  45  degrees  read  down  the  left-hand  side  of  the 
page;  for  angles  between  45  and  90  degrees  read  up  the  right- 
hand  side. 

The  characteristics  of  the  logarithmic  functions  are  printed 
in  full-faced  type  at  or  near  the  top  of  the  columns  headed  Log. 
A  glance  at  the  natural  function  will  determine  whether  or  not 
10  is  to  be  subtracted  from  the  characteristic  given. 

For  reducing  minutes  to  decimal  parts  of  a  degree,  and 
vice  versa,  use  the  table  on  page  40. 


NATURAL   LOGARITHMS 


Natural    Logarithms    of    Numbers    1.00 — 9.99    and    1 — 999 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

1.0 
1.1 
1.2 
1.3 

0.0000 
0.09531 
0.1823 
0.2624 

0.00995 
0.1044 
0.1906 
0.2700 

0.01980 
0.1133 
0.1988 
0.2776 

0.02956 
0.1222 
0.2070 
0.2852 

0.03922 
0.1310 
0.2151 
0.2927 

0.04879 
0.1398 
0.2231 
0.3001 

0.05827 
0.1484 
0.2311 
0.3075 

0.06766 
0.1570 
0.2390 
0.3148 

0.07696 
0.1655 
0.2469 
0.3221 

0.08618 
0.1740 
0.2546 
0.3293 

1.4 
1.5 

1.6 

0.3365 
0.4055 
0.4700 

0.3436 
0.4121 
0.4762 

0.3507 
0.4187 

0.4824 

0.3577 
0.4253 
0.4886 

0.3646 
0.4318 
0.4947 

0.3716 
0.4383 
0.500*, 

0.3784 
0.4447 
0.5068 

0.3853 
0.4511 
0.5128 

0.3920 
0.4574 

0.5188 

0.3988 
0.4637 
0.5247 

1.7 
l.S 
1.9 

0.5306 
0.5878 
0.6419 

0.5365 
0.5933 
0.6471 

0.5423 
0.5988 
0.6523 

0.5481 
0.6043 
0.6575 

0.5539 
0.6098 
0.6627 

0.5596 
0.6152 
0.6678 

0.5653 
0.6206 
0.6729 

0.5710 
0.6259 
0.6780 

0.5766 
0.6313 
0.6831 

0.5822 
0.6366 
0.6881 

2.0 
2.1 
2.2 
2.3 

0.6931 
0.7419 
0.7885 
0.8329 

0.6981 
0.7467 
0.7930 
0.8372 

0.7031 
0.7514 
0.7975 
0.8416 

0.7080 
0.7561 
0.8020 
0.8459 

0.7129 

0.7608 
0.8065 
0.8502 

0.7178 
0.7655 
0.8109 
0.8544 

0.7227 
0.7701 
0.8154 
0.8587 

0.7275 

0.7747 
0.8198 
0.8629 

0.7324 
0.7793 
0.8242 
0.8671 

0.7372 
0.7839 
0.828G 
0.8713 

2.4 
2.5 
2.6 

0.8755 
0.9163 
0.9555 

0.8796 
0.9203 
0.9594 

0.8838 
0.9243 
0.9632 

0.8879 
0.9282 
0.9670 

0.8920 
0.9322 
0.9708 

0.8961 
0.9361 
0.9746 

0.9002 
0.9400 
0.9783 

0.9042 
0.9439 
0.9821 

0.9083 
0.9478 
0.9858 

0.9123 
0.9517 
0.9895 

2.7 
2.8 
2.9 

0.9933 
1.0296 
1.0647 

0.9969 
1.0332 
1.0682 

1.0006 
1.0367 
1.0716 

1.0043 
1.0403 
1.0750 

1.0080 
1.0438 
1.0784 

1.0116 
1.0473 

1.0818 

1.0152 
1.0508 
1.0852 

1.0188 
1.0543 
1.0886 

1.0225 
1.0578 
1.0919 

1. 
1; 
1. 

3.0 

3.1 
3.2 
3.3 

1.0986 
1.1314 
1.1632 
1.1939 

1.1019 
1.1346 
1.1663 
1.1969 

1.1053 
1.1378 
1.1694 
1.2000 

1.1086 
1.1410 
1.1725 
1.2030 

1.1119 
1.1442 
1.1756 
1.2060 

1.1151 
1.1474 

1.1787 
1.2090 

1.1184 
1.1506 
1.1817 
1.2119 

1.1217 
1.1537 
1.1848 
1.2149 

1.1249 
1.1569 
1.1878 
1.2179 

1. 
1. 
1. 
1.1 

3.4 
3.5 
3.6 

1.2238 
1.2528 
1.2809 

1.2267 
1.2556 
1.2837 

1.2296 
1.2585 
1.2865 

1.2326 
1.2613 
1.2892 

1.2355 
1.2641 
1.2920 

1.2384 
1.2669 
1.2947 

1.2413 
1.2698 
1.2975 

1.2442 
i:2726 
1.3002 

1.2470 
1.2754 
1.3029 

i. 

1. 
1. 

3.7 
3.8 
3.9 

1.3083 
1.3350 
1.3610 

1.3110 
1.3376 
1.3635 

1.3137 
1.3403 
1.3661 

1.3164 
1.3429 
1.3686 

1.3191 
1.3455 
1.3712 

1.3218 
1.3481 
1.3737 

1.3244 
1.3507 
1.3762 

1.3271 
1.3533 

1.3788 

1.3297 
1.3558 
1.3813 

1, 

1. 
i.- 

4.0 
4.1 

4.  "s" 

1.3863 
1.4110 
1.4351 
1.4586 

1.3888 
1.4134 
1.4375 
1.4609 

1.3913 
1.4159 
1.4398 
1.4633 

1.3938 
1.4183 
1.4422 
1.4656 

1.3962 
1.4207 
1.4446 
1.4679 

1.3987 
1.4231 
1.4469 
1.4702 

1.4012 
1.4255 
1.4493 
1.4725 

1.4036 
1.4279 
1.4516 
1.4748 

1.4061 
1.4303 
1.4540 
1.4770 

i. 
i. 

i.' 
i. 

4.4 
4.5 
4.6 

1.4816 
1.5041 
1.5261 

1.4839 
1.5063 
1.5282 

1.4861 
1.5085 
1.5304 

1.4884 
1.5107 
1.5326 

1.4907 
1.5129 
1.5347 

1.4929 
1.5151 
1.5369 

1.4951 
1.5173 
1.5390 

1.4974 
1.5195 
1.5412 

1.4996 
1.5217 
1.5433 

1.1 
i. 
i. 

4.7 
4.8 
4.9 

1.5476 
1.5686 
1.5892 

1.5497 
1.5707 
1.5913 

1.5518 
1.5728 
1.5933 

1.5539 

1.5748 
1.5953 

1.5560 
1.5769 
1.5974 

1.5581 
1.5790 
1.5994 

1.5602 
1.5810 
1.6014 

1.5623 
1.5831 
1.6034 

1.5644 
1.5851 
1.6054 

i. 
1.1 

i. 

NATURAL   LOGARITHMS 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

5.0 
5.1 
5.2 
5.3 

1.6094 
1.6292 
1.6487 
1.6677 

1.6114 
1.6312 
1.6506 
1.6696 

1.6134 
1.6332 
1.6525 
1.6715 

1.6154 
1.6351 
1.6544 
1.6734 

1.6174 
1.6371 
1.6563 
1.6752 

1.6194 
1.6390 
1.6582 
1.6771 

1.6214 
1.6409 
1.6601 
1.6790 

1.6233 
1.6429 
1.6620 
1.6808 

1.6253 
1.6448 
1.6639 
1.6827 

1.6273 
1.6467 
1.6658 
1.6845 

5.4 
5.5 
5.6 

1.6864 
1.7047 
1.7228 

1.6882 
1.7066 
1.7246 

1.6901 
1.7084 
1.7263 

1.6919 
1.7102 
1.7281 

1.6938 
1.7120 
1.7299 

1.6956 
1.7138 
1.7317 

1.6974 
1.7156 
1.7334 

1.6993 
1.7174 
1.7352 

1.7011 
1.7192 
1.7370 

1.7029 
1.7210 
1.7387 

5.7 
5.8 
5.9 

1.7405 
1.7579 
1.7750 

1.7422 
1.7596 
1.7766 

1.7440 
1.7613 
1.7783 

1.7457 
1.7630 

1.7800 

1.7475 
1.7647 
1.7817 

1.7492 
1.7664 
1.7834 

1.7509 
1.7681 
1.7851 

1.7527 
1.7699 
1.7867 

1.7544 
1.7716 

1.7884 

1.7561 
1.7733 
1.7901 

6.0 
6.1 
6.2 
6.3 

1.7918 
1.8083 
1.8245 
1.8405 

1.7934 
1.8099 
1.8262 
1.8421 

1.7951 
1.8116 
1.8278 
1.8437 

1.7967 
1.8132 
1.8294 
1.8453 

1.7984 
1.8148 
1.8310 
1.8469 

1.8001 
1.8165 
1.8326 
1.8485 

1.8017 
1.8181 
1.8342 
1.8500 

1.8034 
1.8197 
1.8358 
1.8516 

1.8050 
1.8213 
1.8374 
1.8532 

18.066 
1.8229 
1.8390 
1.8547 

6.4 
6.5 
6.6 

1.8563 
1.8718 

1.8871 

1.8579 
1.8733 
1.8886 

1.8594 
1.8749 
1.8901 

1.8610 
1.8764 
1.8916 

1.8625 
1.8779 
1.8931 

1.8641 
1.8795 
1.8946 

1.8656 
1.8810 
1.8961 

1.8672 
1.8825 
1.8976 

1.8687 
1.8840 
1.8991 

1.8703 
1.8856 
1.9006 

S.7 
6.8 
6.9 

1.9021 
1.9169 
1.9315 

1.9036 
1.9184 
1.9330 

1.9051 
1.9199 
1.9344 

1.9066 
1.9213 
1.9359 

1.9081 
1.9228 
1.9373 

1.9095 
1.9242 
1.9387 

1.9110 
1.9257 
1.9402 

1.9125 
1.9272 
1.9416 

1.9140 
1.9286 
1.9430 

1.9155 
1.9301 
1.9445 

7.0 
7.1 
7.2 
7.3 

1.9459 
1.9601 
1.9741 
1.9879 

1.9473 
1.9615 
1.9755 
1.9892 

1.9488 
1.9629 
1.9769 
1.9906 

1.9502 
1.9643 
1.9782 
1.9920 

1.9516 
1.9657 
1.9796 
1.9933 

1.9530 
1.9671 
1.9810 
1.9947 

1.9544 
1.9685 
1.9824 
1.9961 

1.9559 
1.9699 
1.9838 
1.9974 

1.9573 
1.9713 
1.9851 
1.9988 

1.9587 
1.9727 
1.9865 
2.0001 

M 

7.5 
7.6 

2.0015 
2.0149 

2.0281 

2.0028 
2.0162 
2.0295 

2.0042 
2.0176 
2.0308 

2.0055 
2.0189 
2.0321 

2.0069 

2.0202 
2.0334 

2.0082 
2.0215 
2.0347 

2.0096 
2.0229 
2.0360 

2.0109 
2.0242 
2.0373 

2.0122 
2.0255 
2.0386 

2.0136 
2.0268 
2.0399 

r.7 
r.s 

7.9 

2.0412 
2.0541 
2.0669 

2.0425 
2.0554 
2.0681 

2.0438 
2.0567 
2.0694 

2.0451 
2.0580 
2.0707 

2.0464 
2.0592 
2.0719 

2.0477 
2.0605 
2.0732 

2.0490 
2.0618 
2.0744 

2.0503 
2.0631 
2.0757 

2.0516 
2.0643 
2.0769 

2.0528 
2.0656 
2.0782 

8.0 

5.1 
5.2 
5.3 

2.0794 
2.0919 
2.1041 
2.1163 

2.0807 
2.0931 
2.1054 
2.1175 

2.0819 
2.0943 
2.1066 
2.1187 

2.0832 
2.0956 
2.1078 
2.1199 

2.0844 
2.0968 
2.1090 
2.1211 

2.0857 
2.0980 
2.1102 
2.1223 

2.0869 
2.0992 
2.1114 
2.1235 

2.0882 
2.1005 
2.1126 
2.1247 

2.0894 
2.1017 
2.1138 
2.1258 

2.0906 
2.1029 
2.1150 
2.1270 

5.4 
5.5 

5.6 

2.1282 
2.1401 
2.1518 

2.1294 
2.1412 
2.1529 

2.1306 
2.1424 
2.1541 

2.1318 
2.1436 
2.1552 

2.1330 
2.1448 
2.1564 

2.1342 
2.1459 
2.1576 

2.1353 
2.1471 
2.1587 

2.1365 
2.1483 
2.1599 

2,1377 

2.1494 
2.1610 

2.1389 
2.1506 
2.1622 

5.7 
1.8 
5.9 

2.1633 
2.1748 
2.1861 

2.1645 
2.1759 
2.1872 

2.1656 
2.1770 
2.1883 

2.1668 
2.1782. 
2.1894 

2.1679 
2.1793 
2.1905 

2.1691 
2.1804 
2.1917 

2.1702 
2.1815 
2.1928 

2.1713 
2.1827 
2.1939 

2.1725 
2.1838 
2.1950 

2.1736 
2.1849 
2.1961 

>.o 

).l 
).2 
>.3 

2.1972 
2.2083 
2.2192 
2.2300 

2.1983 
2.2094 
2.2203 
2.2311 

2.1994 
2.2105 
2.2214 
2.2322 

2.2006 
2.2116 
2.2225 
2.2332 

2.2017 
2.2127 
2.2235 
2.2343 

2.2028 
2.2138 
2.2246 
2.2354 

2.2039 
2.2148 
2.2257 
2.2364 

2.2050 
2.2159 
2.2268 
2.2375 

2.2061 
2.2170 
2.2279 
2.2386 

2.2072 
2.2181 
2.2289 
2.2396 

).4 
).5 
).6 

2.2407 
2.2513 
2.2618 

2.2418 
2.2523 
2.2628 

2.2428 
2.2534 
2.2638 

2.2439 
2.2544 
2.2649 

2.2450 
2.2555 
2.2659 

2.2460 
2.2565 
2.2670 

2.2471 
2.2576 
2.2680 

2.2481 
2.2586 
2.2690 

2.2492 
2.2597 
2.2701 

2.2502 
2.2607 
2.2711 

).7 
).8 
1.9 

2.2721 
2.2824 
2.2925 

2.2732 

2.2834 
2.2935 

2.2742 

2.2844 
2.2946 

2.2752 
2.2854 
2.2956 

2.2762 

2.2865 
2.2966 

2.2773 

2.2875 
2.2976 

2.2783 
2.2885 
2.2986 

2.2793 
2.2895 
2.2996 

2.2803 
2.2905 
2.3006 

2.2814 
2.2915 
2.3016 

NATURAL  LOGARITHMS 


N 


8 


9 


10 
11 
12 

13 

14 
15 

16 

17 

18 
19 

20 
21 

22 

23 

24 
25 

26 


38 
39 

40 
41 
42 
43 

44 
45 
46 

47 
48 
49 


0.00000  0.69315  1.09861  1.38629 
2.30259  2.39790  2.48491  2.56495  2.63906 
2.99573  3.04452  3.09104  3.13549  3.17805 
3.40120  3.43399  3.46574  3.49651  3.52636 

3.68888  3.71357  3.73767  3.76120  3.78419 
3.91202  3.93183  3.95124  3.97029  3.98898 
4.09434  4.11087  4.12713  4.14313  4.15888 


,24850  4.26268 
,38203  4.39445 
49981  4.51086 


.27667  4.29046  4.30407 
.40672  4.41884  4.43082 
.52179  4.53260  4.54329 


.60517  4.61512  .62497  4.63473  4.64439 

.70048  4.70953  .71850  4.72739  4.73620 

.78749  4.79579  4.80402  4.81218  4.82028 

4.86753  4.87520  4.88280  4.89035  4.89784 

4.94164  4.94876  4.95583  4.96284  4.96981 
5.01064  5.01728  5.02388  5.03044  5.03695 
5.07517  5.08140  5.08760  5.09375  5.09987 

5.13580  5.14166  5.14749  5.15329  5.15906 
5.19296  5.19850  5.20401  5.20949  5.21494 
5.24702  5.25227  5.25750  5.26269  5.26786 

5.29832  5.30330  5.30827  5.31321  5.31812 
5.34711  5.35186  5.35659  5.36129  5.36598 
5.39363  5.39816  5.40268  5.40717  5.41165 
5.43808  5.44242  5.44674  5.45104  5.45532 

5.48064  5.48480  5.48894  5.49306  5.49717 
5.52146  5.52545  5.52943  5.53339  5.53733 
5.56068  5.56452  5.56834  5.57215  5.57595 

5.59842  5.60212  5.60580  5.60947  5.61313 
5.63479  5.63835  5.64191  5.64545  5.64897 
5.66988  5.67332  5.67675  5.68017  5.68358 

5.70378  5.70711  5.71043  5.71373  5.71703 
5.73657  5.73979  5.74300  5.74620  5.74939 
5.76832  5.77144  5.77455  5.77765  5.78074 
5.79909  5.80212  5.80513  5.80814  5.81114 

5.82895  5.83188  5.83481  5.83773  5.84064 
5.85793  5.86079  5.86363  5.86647  5.86930 
5.88610  5.88888  5.89164  5.89440  5.89715 

5.91350  5.91620  5.91889  5.92158  5.92426 
5.94017  5.94280  5.94542  5.94803  5.95064 
5.96615  5.96871  5.97126  5.97381  5.97635 

5.99146  5.99396  5.99645  5.99894  6.00141 

6.01616  6.01859  6.02102  6.02345  6.02587 

6.04025  6.04263  6.04501  6.04737  6.04973 

6.06379  6.06611  6.06843  6.07074  6.07304 

6.08677  6.08904  6.09131  6.09357  6.09582 
6.10925  6.11147  6.11368  6.11589  6.11810 
6.13123  6.13340  6.13556  6.13773  6.13988 

6.15273  6.15486  6.15698  6.15910  6.16121 
6.17379  6.17587  6.17794  6.18002  6.18208 
6.19441  6.19644  6.19848  6.20051  6.20254 


1.60944  1.79176  1.94591  2.07944  2.19722 
2.70805  2.77259  2.83321  2.89037  2.94444 
3.21888  3.25810  3.29584  3.33220  3.36730 
3.55535  3.58352  3.61092  3.63759  3.66356 

3.80666  3.82864  3.85015  3.87120  3.89182 

4.00733  4.02535  4.04305  4.06044  4.07754 

4.17439  4.18965  4.20469  4.21951  4.23411 

4.31749  4.33073  4.34381  4.35671  4.36945 
4.44265  4.45435  4.46591  4.47734  4.48864 

4.55388  4.56435  4.57471  4.58497  4.59512 

4.65396  4.66344  4.67283  4.68213  4.69135 

4.74493  4.75359  4.76217  4.77068  4.77912 

4.82831  4.83628  4.84419  4.85203  4.85981 

4.90527  4.91265  4.91998  4.92725  4.93447 

4.97673  4.98361  4.99043  4.99721  5.00395 

5.04343  5.04986  5.05625  5.06260  5.06890 

5.10595  5.11199  5.11799  5.12396  5.12990 

5.16479  5.17048  5.17615  5.18178  5.19739 

5.22036  5.22575  5.23111  5.23644  5.24175 

5.27300  5.27811  5.28320  5.28827  5.29330 

5.32301  5.32788  5.33272  5.33754  5.34233 

5.37064  5.37528  5.37990  5.38450  5.38907 

5.41610  5.42053  5.42495  5.42935  5.43372 

5.45959  5.46383  5.46806  5.4.7227  5.47646 

5.50126  5.50533  5.50939  5.51343  5.51745 

5.54126  5.54518  5.54908  5.55296  5.55683 

5.57973  5.58350  5.58725  5.59099  5.59471 

5.61677  5.62040  5.62402  5.62762  5.63121 

5.65249  5.65599  5.65948  5.66296  5.66643 

5.68698  5.69036  5.69373  5.69709  5.70044 

5.72031  5.72359  5.72685  5.73010  5.73334 

5.75257  5.75574  5.75890  5.76205  5.76519 

5.78383  5.78690  5.78996  5.79301  5.79606 

5.81413  5.81711  5.82008  5.82305  5.82600 

5.84354  5.84644  5.84932  5.85220  5.85507 

5.87212  5.87493  5.87774  5.88053  5.88332 

5.89990  5.90263  5.90536  5.90808  5.91080 

5.92693  5.92959  5.93225  5.93489  5.93754 

5.95324  5.95584  5.95842  5.96101  5.96358 

5.97889  5.98141  5.98394  5.98645  5.98896 

6.00389  6.00635  6.00881  6.01127  6.01372 

6.02828  6.03069  6.03309  6.03548  6.03787 

6.05209  6.05444  6.05678  6.05912  6.06146 

6.07535  6.07764  6.07993  6.08222  6.08450 

6.09807  6.10032  6.10256  6.10479  6.10702 

6.12030  6.12249  6.12468  6.12687  6.12905 

6.14204  6.14419  6.14633  6.14847  6.15060 

6.16331  6.16542  6.16752  6.16961  6.17170 

6.18415  6.18621  6.18826  6.19032  6.19236 

6.20456  6.20658  6.20859  6.21060  6.21261 


6 


9 


NATURAL  LOGARITHMS 


3 


50  6.21461  6.21661  6.21860  6.22059  6.22258 

51  6.23441  6.23637  6.23832  6.24028  6.24222 

52  6.25383  6.25575  6.25767  6.25958  6.26149 

53  6.27288  6.27476  6.27664  6.27852  6.28040 

54  6.29157  6.29342  6.29527  6.29711  6.29895 

55  6.30992  6.31173  6.31355  6.31536  6.31716 

56  6.32794  6.32972  6.33150  6.33328  6.33505 

57  6.34564  6.34739  6.34914  6.35089  6.35263 

58  6.36303  6.36475  6.36647  6.36819  6.36990 

59  6.38012  6.38182  6.38351  6.38519  6.38688 

60  6.39693  6.39859  6.40026  6.40192  6.40357 

61  6.41346  6.41510  6.41673  6.41836  6.41999 

62  6.42972  6.43133  6.43294  6.43455  6.43615 

63  6.44572  6.44731  6.44889  6.45047  6.45205 

64  6.46147  6.46303  6.46459  6.46614  6.46770 

65  6.47697  6.47851  6.48004  6.48158  6.48311 

66  6.49224  6.49375  6.49527  6.49677  6.49828 

67  6.50728  6.50877  6.51026  6.51175  6.51323 

68  6.52209  6.52356  6.52503  6.52649  6.52796 

69  6.53669  6.53814  6.53959  6.54103  6.54247 

70  6.55108  6.55251  6.55393  6.55536  6.55678 

71  6.56526  6.56667  6.56808  6.56948  6.57088 

72  6.57925  6.58064  6.58203  6.58341  6.58479 

73  6.59304  6.59441  6.59578  6.59715  6.59851 

74  6.60665  6.60800  6.60935  6.61070  6.61204 

75  6.62007  6.62141  6.62274  6.62407  6.62539 

76  6.63332  6.63463  6.63595  6.63726  6.63857 

77  6.64639  6.64769  6.64898  6.65028  6.65157 

78  6.65929  6.66058  6.66185  6.66313  6.66441 

79  6.67203  6.67330  6.67456  6.67582  6.67708 

80  6.68461  6.68586  6.68711  6.6S835  6.68960 

81  6.69703  6.69827  6.69950  6.70073  6.70196 

82  6.70930  6.71052  6.71174  6.71296  6.71417 

83  6.72143  6.72263  6.72383  6.72503  6.72623 

84  6.73340  6.73459  6.73578  "6.73697  6.73815 

85  6.74524  6.74641  6.74759  6.74876  6.74993 

86  6.75693  6.75809  6.75926  6.76041  6.76157 

87  6.76849  6.76964  6.77079  6.77194  6.77308 

88  6.77992  6.78106  6.78219  6.78333  6.78446 

89  6.79122  6.79234  6.79347  6.79459  6.79571 

90  6.80239  6.80351  6.80461  6.80572  6.80683 

91  6.81344  6.81454  6.81564  6.81674  6.81783 

92  6.82437  6.82546  6.82655  6.82763  6.82871 

93  6.83518  6.83626  6.83733  6.83841  6.83948 

94  6.84588  6.84694  6.84801  6.84907  6.85013 

95  6.85646  6.85751  6.85857  6.85961  6.86066 

96  6.86693  6.86797  6.86901  6.87005  6.87109 

97  6.87730  6.87833  6.87936  6.88038  6.88141 

98  6.88755  6.88857  6.88959  6.89061  6.89163 

99  6.89770  6.89871  6.89972  6.90073  6.90174 


6.22456  6.22654  6.22851  6.23048  6.23245 

6.24417  6.24611  6.24804  6.24998  6.25190 

6.26340  6.26530  6.26720  6.26910  6.27099 

6.28227  6.28413  6.28600  6.28786  6.28972 

6.30079  6.30262  6.30445  6.30628  6.30810 

6.31897  6.32077  6.32257  6.32436  6.32615 

6.33683  6.33859  6.34036  6.34212  6.34388 

6.35437  6.35611  6.35784  6.35957  6.36130 

6.37161  6.37332  6.37502  6.37673  6.37843 

6.38856  6.39024  6.39192  6.39359  6.39526 

6.40523  6.40688  6.40853  6.41017  6.41182 

6.42162  6.42325  6.42487  6.42649  6.42811 

6.43775  6.43935  6.44095  6.44254  6.44413 

6.45362  6.45520  6.45677  6.45834  6.45990 

6.46925  6.47080  6.47235  6.47389  6.47543 

6.48464  6.48616  6.48768  6.48920  6.49072 

6.49979  6.50129  6.50279  6.50429  6.50578 

6.51471  6.51619  6.51767  6.51915  6.52062 

6.52942  6.53088  6.53233  6.53379  6.53524 

6.54391  6.54535  6.54679  6.54822  6.54965 

6.55820  6.55962  6.56103  6.56244  6.56386 

6.57228  6.57368  6.57508  6.57647  6.57786 

6.58617  6.58755  6.58893  6.59030  6.59167 

6.59987  6.60123  6.60259  6.60394  6.60530 

6.61338  6.61473  6.61607  6.61740  6.61874 

6.62672  6.62804  6.62936  6.63068  6.63200 

6.63988  6.64118  6.64249  6.64379  6.64509 

6.65286  6.65415  6.65544  6.65673  6.65801 

6.66568  6.66696  6.66823  6.66950  6.67077 

6.67834  6.67960  6.68085  6.68211  6.68336 

6.69084  6.69208  6.69332  6.69456  6.69580 
6.70319  6.70441  6.70564  6.70686  6.70808 

6.71538  6.71659  6.71780  6.71901  6.72022 

6.72743  6.72863  6.72982  6.73102  6.73221 

6.73934  6.74052  6.74170  6.74288  6.74406 

6.75110  6.75227  6.75344  6.75460  6.75577 

6.76273  6.76388  6.76504  6.76619  6.76734 

6.77422  6.77537  6.77651  6.77765  6.77878 

6.78559  6.78672  6.78784  6.78897  6.79010 

6.79682  6.79794  6.79906  6.80017  6.80128 

6.80793  6.80904  6.81014  6.81124  6.81235 

6.81892  6.82002  6.82111  6.82220  6.82329 

6.82979  6.83087  6.83195  6.83303  6.83411 

6.84055  6.84162  6.84268  6.84375  6.84482 

6.85118  6.85224  6.85330  6.85435  6.85541 

6.86171  6.86276  6.86380  6.86485  6.86589 

6.87213  6.87316  6.87420  6.87523  6.87626 

6.88244  6.88346  6.88449  6.88551  6.88653 
6.89264  6.89366  6.89467  6.89568  6.89669 

6.90274  6.90375  6.90475  6.90575  6.90675 


6 


8 


9 


10 


COMMON   LOGARITHMS 


Logarithms  of  Numbers  1 — 100 


N-        Log. 


N       Log. 


N        Log. 


N        Log. 


N        Log. 


1 

0.00000 

21 

1 

.32222 

41 

1 

.61278 

61 

1.78533 

81 

1.90849 

2 

0.30103 

22 

1 

.34242 

42 

1 

.62325 

62 

1.79239 

82 

1.91381 

3 

0.47712 

23 

1 

.36173 

43 

1 

.63347 

63 

1.79934 

83 

1.91908 

4 

0.60206 

24 

1.38021 

44 

1 

64345 

64 

1.80618 

84 

1.  92428 

5 

0.69897 

25 

1 

.39794 

45 

1 

.65321 

65 

1.81291 

85 

1.92942 

6 

0.77815 

26 

1 

.41497 

46 

1 

.66276 

66 

1.81954 

86 

1.93450 

7 

0.84510 

27 

1 

.43136 

47 

1 

.67210 

67 

.82607 

87 

1.93952 

8 

0.90309 

28 

1 

.44716 

48 

1 

.68124 

68 

.83251 

88 

1.94448 

9 

0.95424 

29 

1 

.46240 

49 

1 

,69020 

69 

.83885 

89 

1.94939 

10 

1.00000 

30 

1 

,47712 

50 

1 

.69897 

70 

.84510 

90 

1.95424 

11 

1.04139 

31 

1 

.49136 

51 

1 

.70757 

71 

.85126 

91 

1.95904 

12 

1.07918 

32 

1 

.50515 

52 

1 

.71600 

72 

1.85733 

92 

1.96379 

13 

1.11394 

33 

1 

.51851 

53 

1 

.72428 

73 

1.86332 

93 

1.96848 

14 

1.14613 

34 

1 

.53148 

54 

1 

.73239 

74 

1.86923 

94 

1.97313 

15 

1.17609 

35 

1 

.54407 

55 

1 

.74036 

75 

1.87506 

95 

1.97772 

16 

1.20412 

36 

1 

.55630 

56 

1 

.74819 

76 

1.88081 

96 

1.98227 

17 

1.23045 

37 

1 

.56820 

57 

1 

.75587 

77 

1.88649 

97 

1.98677 

18 

1.25527 

38 

1 

.57978 

58 

1 

.76343 

78 

1.89209 

98. 

1.99123 

19 

1.27875 

39 

1 

.59106 

59 

1 

.77085 

79 

1.89763 

99 

1.99564 

20 

1.30103 

40 

1 

.60206 

60 

1 

.77815 

80 

1.90309 

100 

2.000UO 

Mantissas    of    Numbers    100—10000 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 

100 

101 

00000 
00432 

00043 
00475 

00087 
00518 

00130 
00561 

00173 
00604 

00217 
00647 

00260 
00689 

00303 
00732 

00346 
00775 

00389 
00817 

43 

1Q2 
103 
104 

00860 
01284 
01703 

00903 
01326 
01745 

00945 
01368 
01787 

00988 
01410 
01828 

01030 
01452 
01870 

01072 
01494 
01912 

01115 
01536 
01953 

01157 
01578 
01995 

01199 
01620 
02036 

01242 
01662 
02078 

42 

105 

106 

02119 
02531 

02160 
02572 

02202 
02612 

02243 
02653 

02284 
02694 

02325 
02735 

02366 
02776 

02407 
02816 

02449 

02857 

02490 
02898 

41 

107  02938    02979    03019    03060    03100 

108  03342    03383    03423    03463    03503 

109  03743    03782    03822    03862    03902 


03141    03181    03222    03262    03302 
03543    03583    03623    03663    03703    40 
03941    03981    04021    04060    04100 


Proportional  Parts 


D 

1 

2 

3 

4 

5 

6 

7 

8 

9 

44 

4.4 

8.8 

13.2 

17.6 

22.0 

26.4 

30.8 

35.2 

39.6 

43 

4.3 

8.6 

12.9 

17.2 

21.5 

25.8 

30.1 

34.4 

38.7 

42 

4.2 

8.4 

12.6 

16.8 

21.0 

25.2 

29.4 

33.6 

37.8 

41 

4.1 

8.2 

12.3 

16.4 

20.5 

24.6 

28.7 

32.8 

36.9 

40 

4.0 

8.0 

12.0 

16.0 

20.0 

24.0 

28.0 

32.0 

36.0 

COMMON   LOGARITHMS 


11 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 

110 
111 
112 

04139 
04532 
04922 

04179 
04571 
04961 

04218 
04610 
04999 

04258 
04650 
05038 

04297 
04689 
05077 

04336 

04727 
05115 

04376 
04766 
05154 

04415 
04805 
05192 

04454 
04844 
05231 

04493 
04883 
05269 

39 

113 
114 

115 

05308 
05690 

06070 

05346 
05729 

06108 

05385 
05767 

06145 

05423 
05805 

06183 

05461 
05843 

06221 

05500 
05881 

06258 

05538 
05918 

06296 

05576 
05956 

06333 

05614 
05994 

06371 

05652 
06032 

06408 

38 

116 
117 
118 

06446 
06819 
07188 

06483 
06,856 
07225 

06521 
06893 
07262 

06558 
06930 
07298 

06595 
06967 
07335 

06633 
07004 
07372 

06670 
07041 
07408 

06707 
07078 
07445 

06744 
07115 
07482 

06781 
07151 
07518 

37 

119 

120 
121 

07555 

07918 

08279 

07591 

07954 
08314 

07628 

07990 
08350 

07664 

08027 
08386 

07700 

08063 
08422 

07737 

08099 
08458 

07773 

08135 
08493 

07809 

08171 
08529 

07846 

08207 
08565 

07882 

08243 
08600 

36 

122 

123 
124 

125 

08636 
08991 
09342 

09691 

08672 
09026 
09377 

09726 

08707 
09061 
09412 

09760 

08743 
09096 
09447 

09795 

08778 
09132 
09482 

09830 

08814 
09167 
09517 

09864 

08849 
09202 
09552 

09899 

08884 
09237 
09587 

09934 

08920 
09272 
09621 

09968 

08955 
09307 
09656 

10003 

35 

126 
127 
128 
129 

10037 
10380 
10721 
11059 

10072 
10415 
10755 
11093 

10106 
10449 
10789 
11126 

10140 
10483 
10823 
11160 

10175 
10517 
10857 
11193 

10209 
10551 
10890 
11227 

10243 
10585 
10924 
11261 

10278 
10619 
10958 
11294 

10312 
10653 
10992 
11327 

10346 
10687 
11025 
11361 

34 

130 
131 
132 
133 

11394 
11727 
12057 
12385 

11428 
11760 
12090 
12418 

11461 
11793 
12123 
12450 

11494 
11826 
12156 
12483 

11528 
11860 
12189 
12516 

11561 
11893 
12222 
12548 

11594 
11926 
12254 
12581 

11628 
11959 
12287 
12613 

11661 
11992 
12320 
12646 

11694 
12024 
12352 
12678 

33 

134 

135 
136 
137 

12710 

13033 
13354 
13672 

12743 

13066 
13386 
13704 

12775 

13098 
13418 
13735 

12808 

13130 
13450 
13767 

12840 

13162 
13481 
13799 

12872 

13194 
13513 
13830 

12905 

13226 
13545 
13862 

12937 

13258 
13577 
13893 

12969 

13290 
13609 
13925 

13001 

13322 
13640 
13956 

32 

138 
139 

13988 
14301 

14019 
14333 

14051 
14364 

14082 
14395 

14114 
14426 

14145 
14457 

14176 
14489 

14208 
14520 

14239 
14551 

14270 
14582 

31 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 

Proportional    Parts 


D 

1 

2 

3 

4 

5 

6 

7 

8 

9 

39 

3.9 

7.8 

11.7 

15.6 

19.5 

23.4 

27.3 

31.2 

35.1 

38 

3.8 

7.6 

11.4 

15.2 

19.0 

22.8 

26.6 

30.4 

34.2 

37 

3.7 

7.4 

11.1 

14.8 

18.5 

22.2 

25.9 

29.6 

33.3 

36 

3.6 

7.2 

10.8 

14.4 

18.0 

21.6 

25.2 

28.8 

32.4 

35 

3.5 

7.0 

10.5 

14.0 

17.5 

21.0 

24.5 

28.0 

31.5 

34 

3.4 

6.8 

10.2 

13.6 

17.0 

20.4 

23.8 

27.2 

30.6 

33 

3.3 

6.6 

9.9 

13.2 

16.5 

19.8 

23.1 

26.4 

29.7 

32 

3.2 

6.4 

9.6 

12.8 

16.0 

19.2 

22.4 

25.6 

28.8 

31 

3.1 

6.2 

9.3 

12.4 

15.5 

18.6 

21.7 

24.8 

27.9 

12 


COMMON    LOGARITHMS 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 

140 
141 
142 

14613 

14922 
15229 

14644 
14953 
15259 

14675 
14983 
15290 

14706 
15014 
15320 

14737 
15045 
15351 

14768 
15076 
15381 

14799 
15106 
15412 

14829 
15137 
15442 

14860 
15168 
15473 

14891 
15198 
15503 

31 

143 
144 

145 
146 

15534 
15836 

16137 
16435 

15564 
15866 

16167 
16465 

15594 
15897 

16197 
16495 

15625 
15927 

16227 
16524 

15655 
15957 

16256 
16554 

15685 
15987 

16286 
16584 

15715 
16017 

16316 
16613 

15746 
16047 

16346 
16643 

15776 

16077 

16376 
16673 

15806 
16107 

16406 
16702 

30 

147 
148 
149 

16732 
17026 
17319 

16761 
17056 
17348 

16791 
17085 
17377 

16820 
17114 
17406 

16850 
17143 
17435 

16879 
17173 
17464 

16909 
17202 
17493 

16938 
17231 
17522 

16967 
17260 
17551 

16997 
17289 
17580 

nq 

150 

151 
152 

17609 
17898 
18184 

17638 
17926 
18213 

17667 
17955 
18241 

17696 
17984 
18270 

17725 
18013 
18298 

17754 
18041 
18327 

17782 
18070 
18355 

17811 
18099 
18384 

17840 
18127 
18412 

17869 
18156 
18441 

153 
154 

18469 
18752 

18498 
18780 

18526 
18808 

18554 
18837 

18583 
18865 

18611 
18893 

18639 
18921 

18667 
18949 

18696 
18977 

18724 
19005 

155 

156 
157 

19033 
19312 
19590 

19061 
19340 
19618 

19089 
19368 
19645 

19117 
19396 
19673 

19145 
19424 
19700 

19173 
19451 
19728 

19201 
19479 
19756 

19229 
19507 
19783 

19257 
19535 
19811 

19285 
19562 
19838 

28 

158 
159 

19866 
20140 

19893 
20167 

19921 
20194 

19948 
20222 

19976 
20249 

20003 
20276 

20030 
20303 

20058 
20330 

20085 
20358 

20112 
20385 

160 

161 
162 
163 

20412 
20683 
20952 
21219 

20439 
20710 
20978 
21245 

20466 
20737 
21005 
21272 

20493 
20763 
21032 
21299 

20520 
20790 
21059 
21325 

20548 
20817 
21085 
21352 

20575 
20844 
21112 
21378 

20602 
20871 
21139 
21405 

20629 
20898 
21165 
21431 

20656 
20925 
21192 
21458 

27 

164 
165 
166 
167 
168 
169 

21484 
21748 
22011 
22272 
22531 
22789 

21511 
21775 
22037 
22298 
22557 
22814 

21537 
21801 
22063 
22324 
22583 
22840 

21564 
21827 
22089 
22350 
22608 
22866 

21590 
21854 
22115 
22376 
22634 
22891 

21617 
21880 
22141 
22401 
22660 
22917 

21643 
21906 
22167 
22427 
22686 
22943 

21669 
21932 
22194 
22453 
22712 
22968 

21696 
21958 
^2220 
22479 
22737 
22994 

21722 
21985 
22246 
22505 
22763 
23019 

26 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 

Proportional    Parts 


D 

1 

2 

3 

4 

5 

6 

7 

8 

9 

31 

3.1 

6.2 

9.3 

12.4 

15.5 

18.6 

21.7 

24.8 

27.9 

30 

3.0 

6.0 

9.0 

12.0 

15.0 

18.0 

21.0 

24.0 

27.0 

29 

2,9 

5.8 

8.7 

11.6 

14.5 

17,4 

20.3 

23.2 

26.1 

28 

2.8 

5.6 

8.4 

11.2 

14.0 

16.8 

19.6 

22.4 

25.2 

27 

2.7 

5.4 

8.1 

10.8 

13.5 

16.2 

18.9 

21.6 

24.3 

26 

2.6 

5.2 

7.8 

10.4 

13.0 

15.6 

18.2 

20.8 

23.4 

COMMON    LOGARITHMS  13 


N01234      56789D 

170  23045  23070  23096  23121  23147  23172  23198  23223  23249  23274 

171  23300  23325  23350  23376  23401  23426  23452  23477  23502  23528 

172  23553  23578  23603  23629  23654  23679  23704  23729  23754  23779 

173  23805  23830  23855  23880  23905  23930  23955  23980  24005  24030 

174  24055  24080  24105  24130  24155  24180  24204  24229  24254  24279 

175  24304  24329  24353  24378  24403  24428  24452  24477  24502  24527 

176  24551  24576  24601  24625  24650  24674  246J9  24724  24748  24773 

177  24797  24822  24846  24871  24895  24920  24944  24969  24993  25018   . 

178  25042  25066  25091  25115  25139  25164  25188  25212  25237  25261 

179  25285  25310  25334  25358  25382  25406  25431  25455  25479  25503 

180  25527  25551  25575  25600  25624  25648  25672  25696  25720  25744 

181  25768  25792  25816  25840  25864  25888  25912  25935  25959  25983  24 

182  26007  26031  26055  26079  26102  26126  26150  26174  26198  26221 

183  26245  26269  26293  26316  26340  26364  26387  26411  26435  26458 

184  26482  26505  26529  26553  26576  26600  26623  26647  26670  26694 

185  26717  26741  26764  26788  26811  26834  26858  26881  26905  26928 

186  26951  26975  26998  27021  27045  27068  27091  27114  27138  27161 

187  27184  27207  27231  27254  27277  27300  27323  27346  27370  27393 

188  27416  27439  27462  27485  27508  27531  27554  27577  27600  27623 

189  27646  27669  27692  27715  27738  27761  27784  27807  27830  27852  23 

190  27875  27898  27921  27944  2796-7  27989  28012  28035  28058  28081 

191  28103  «8126  28149  28171  28194  28217  28240  28262  28285  28307 

192  28330  28353  28375  28398  28421  28443  28466  28488  28511  28533 

193  28556  28578  28601  28623  28646  28668  28691  28713  28735  28758 

194  28780  28803  28825  28847  28870  28892  28914  28937  28959  28981 

195  29003  29026  29048  29070  29092  29115  29137  29159  29181  29203 

196  29226  29248  29270  29292  29314  29336  29358  29380  29403  29425 

197  29447  29469  29491  29513  29535  29557  29579  29601  29623  29645  22 

198  29667  29688  29710  29732  29754  29776  29798  29820  29842  29863 

199  29885  29907  29929  29951  29973  29994  30016  30038  30060  30081 

200  30103  30125  30146  30168  30190  30211  30233  30255  30276  30298 

201  30320  30341  30363  30384  30406  30428  30449  30471  30492  30514 


202  30535  30557  30578  30600  30621  30643  30664  30685  30707  30728 

203  30750  30771  30792  30814  30835  30856  30878  30899  30920  30942  21 

204  30963  30984  31006  31027  31048  31069  31091  31112  31133  31154 


N01234      5678 


Proportional    Parts 


D 

1 

2 

3 

4 

5 

6 

7 

8 

9 

25 

2.5 

5.0 

7.5 

10.0 

12.5 

15.0 

17.5 

20.0 

22.5 

24 

2*4 

4.8 

7.2 

9.6 

12.0 

14.4 

16.8 

19.2 

21.6 

23 

2.3 

4.6 

6.9 

9.2 

11.5 

13.8 

16.1 

18.4 

20.7 

22 

2.2 

4.4 

6.6 

8.8 

11.0 

13.2 

15.4 

17.6 

19.8 

21 

2.1 

4.2 

6.3 

8.4 

10.5 

12.6 

14.7 

16.8 

18.9 

14 


COMMON    LOGARITHMS 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 

205 

31175 

31197 

31218 

31239 

31260 

31281 

31302 

31323 

31345 

31366 

206 

31387 

31408 

31429 

31450 

31471 

31492 

31513 

31534 

31555 

31576 

207 

31597 

31618 

31639 

31660 

31681 

31702 

31723 

31744 

31765 

31785 

21 

208 

31806 

31827 

31848 

31869 

31890 

31911 

31931 

31952 

31973 

31994 

209 

32015 

32035 

32056 

32077 

32098 

32118 

32139 

32160 

32181 

32201 

210 

32222 

32243 

32263 

32284 

32305 

32325 

32346 

32366 

32387 

32408 

211 

32428 

32449 

32469 

32490 

32510 

32531 

32552 

32572 

32593 

32613 

212 

32634 

32654 

32675 

32695 

32715 

32736 

32756 

32777 

32797 

32818 

213 

32838 

32858 

32879 

32899 

32919 

32940 

32960 

32980 

33001 

33021 

214 

33041 

33062 

33082 

33102 

33122 

33143 

33163 

33183 

33203 

33224 

215 

33244 

33264 

33284 

33304 

33325 

33345 

33365 

33385 

33405 

33425 

216 

33445 

33465 

33486 

33506 

33526 

33546 

33566 

33586 

33606 

33626 

20 

217 

33646 

33666 

33686 

33706 

33726 

33746 

33766 

33786 

33806 

33826 

218 

33846 

33866 

33885 

33905 

33925 

33945 

33965 

33985 

34005 

34025 

219 

34044 

34064 

34084 

34104 

34124 

34143 

34163 

34183 

34203 

34223 

220 

34242 

34262 

34282 

34301 

34321 

34341 

34361 

34380 

34400 

34420 

221 

34439 

34459 

34479 

34498 

34518 

34537 

34557 

34577 

34596 

34616 

222 

34635 

34655 

34674 

34694 

34713 

34733 

34753 

34772 

34792 

34811 

223 

34830 

34850 

34869 

34889 

34908 

34928 

34947 

34967 

34986 

35005 

224 

35025 

35044 

35064 

35083 

35102 

35122 

35141 

35160 

35180 

35199 

225 

35218 

35238 

35257 

35276 

35295 

35315 

35334 

35353 

35372 

35392 

226 

35411 

35430 

35449 

35468 

35488 

35507 

35526 

35545 

35564 

35583 

227 

35603 

35622 

35641 

35660 

35679 

35698 

35717 

35736 

35755 

35774 

1  9 

228 

35793 

35813 

35832 

35851 

35870 

35889 

35S08 

35927 

35946 

35965 

Xj 

229 

35984 

36003 

36021 

36040 

36059 

36078 

36097 

36116 

36135 

36154 

230 

36173 

36192 

36211 

36229 

36248 

36267 

36286 

36305 

36324 

36342 

231 

36361 

36380 

36399 

36418 

36436 

36455 

36474 

36493 

36511 

36530 

232 

36549 

36568 

36586 

36605 

36624 

36642 

36661 

36680 

36698 

36717 

233 

36736 

36754 

36773 

36791 

36810 

36829 

36847 

36866 

36884 

36903 

234 

36922 

36940 

36959 

36977 

36996 

37014 

37033 

37051 

37070 

37088 

235 

37107 

37125 

37144 

37162 

37181 

37199 

37218 

37236 

37254 

37273 

236 

37291 

37310 

37328 

37346 

37365 

37383 

37401 

37420 

37438 

37457 

18 

237 

37475 

37493 

37511 

37530 

37548 

37566 

37585 

37603 

37621 

37639 

238 

37658 

37676 

37694 

37712 

37731 

37749 

37767 

37785 

37803 

37822 

239 

37840 

37858 

37876 

37894 

37912 

37931 

37949 

37967 

37985 

38003 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 

Proportional    Parts 


D 

1 

2 

3 

4 

5 

6 

7 

8 

9 

21 

2.1 

4.2 

6.3 

8.4 

10.5 

12.6 

14.7 

16.8 

18.9 

20 

2.0 

4.0 

6.0 

8.0 

10.0 

12.0 

14.0 

16.0 

18.0 

19 

1.9 

3.8 

5.7 

7.6 

9.5 

11.4 

13.3 

15.2 

17.1 

18 

1.8 

3.6 

5.4 

7.2 

9.0 

10.8 

12.6 

14.4 

16.2 

COMMON   LOGARITHMS 


15 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 

240 
241 

242 

38021 
38202 
38382 

38039 
38220 
38399 

38057 
38238 
38417 

38075 
38256 
38435 

38093 
38274 
38453 

38112 
38292 
38471 

38130 
38310 

38489 

38148 
38328 
38507 

38166 
38346 
38525 

38184 
38364 
38543 

243 
244 
245 

38561 
38739 
38917 

38578 
38757 
38934 

38596 

38775 
38952 

38614 
38792 
38970 

38632 

38810 
38987 

38650 
38828 
39005 

38668 
38846 
39023 

38686 
38863 
39041 

38703 
38881 
39058 

38721 
38899 
39076 

18 

246 
247 

248 

39094 
39270 
39445 

39111 
39287 
39463 

39129 
39305 
39480 

39146 
39322 
39498 

39164 
39340 
39515 

39182 
39358 
39533 

39199 
39375 
39550 

39217 
39393 
39568 

39235 
39410 
39585 

39252 
39428 
39602 

249 
250 

251 
252 

39620 
39794 
39967 
40140 

39637 
39811 
39985 
40157 

39655 
39829 
40002 
40175 

39672 
39846 
40019 
40192 

39690 
39863 
40037 
40209 

39707 
39881 
40054 
40226 

39724 
39898 
40071 
40243 

39742 
39915 
40088 
40261 

39759 
39933 
40106 
40278 

39777 
39950 
40123 
40295 

253 
£54 
255 

40312 
40483 
40654 

40329 
40500 
40671 

40346 
40518 
40688 

40364 
40535 
40705 

40381 
40552 
40722 

40398 
40569 
40739 

40415 
40586 
40756 

40432 
40603 
40773 

40449 
40620 
40790 

40466 
40637 
40807 

17 

256 

257 
258 

40824 
40993 
41162 

40841 
41010 
41179 

40858 
41027 
41196 

40875 
41044 
41212 

40892 
41061 
41229 

40909 
41078 
41246 

40926 
41095 
41263 

40943 
41111 
41280 

40960 
41128 
41296 

40976 
41145 
41313 

259 
260 

261 

41330 
41497 
41664 

41347 
41514 
41681 

41363 
41531 
41697 

41380 
41547 
41714 

41397 
41564 
41731 

41414 
41581 

41747 

41430 
41597 
41764 

41447 
41614 
41780 

41464 
41631 
41797 

41481 
41641 
41814 

262 
263 
264 
265 

41830 
41996 
42160 
42325 

41847 
42012 
42177 
42341 

41863 
42029 
42193 
42357 

41880 
42045 
42210 
42374 

41896 
42062 
42226 
42390 

41913 
42078 
42243 
42406 

41929 
42095 
4-J259 
42423 

41946 
42111 
42275 
42439 

41963 
42127 
42292 
42455 

41979 
42144 
42308 
42472 

266 
267 

269 

42488 
42651 
42813 
42975 

42504 
42667 
42830 
42991 

42521 
42684 
42846 
43008 

42537 
42700 
42862 
43024 

42553' 
42716 
42878 
43040 

42570 
42732 
42894 
43056 

42586 
42749 
42911 
43072 

42602 
42765 

42927 
43088 

42619 
42781 
42943 
43104 

42635 
42797 
42959 
43120 

270 

271 
272 
273 

43136 
43297 
43457 
43616 

43152 
43313 
43473 
43632 

43169 
43329 
43489 
43648 

43185 
43345 
43505 
43664 

43201 
43361 
43521 
43680 

43217 
43377 
43537 
43696 

43233 
43393 
43553 
43712 

43249 
43409 
43569 
43727 

43265 
43425 
43584 
43743 

43281 
43441 
43600 
43759 

16 

274 
275 

276 

43775 
43933 
44091 

43791 
43949 
4*107 

43807 
43965 
44122 

43823 
43981 
44138 

43838 
43996 
44154 

43854 
44012 
44170 

43870 
44028 
44185 

43886 
44044 
44201 

43902 
44059 
44217 

43917 
44075 
44232 

277 
279 

44248 
44404 
44560 

44264 
44420 
44576 

44279 
44436 
44592 

44295 
44451 
44607 

44311 
44467 
44623 

44326 
44483 
44638 

44342 
44498 
44654 

44358 
44514 
44669 

44373 
44529 
44685 

44389 
44545 
44700 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 

Proportional  Parts 

D 

1 

2 

3 

4 

5 

6 

7 

8 

9 

18 
17 
16 

1.8 
1.7 
1.6 

3.6 
3.4 
3.2 

5 
5 
4. 

.4     7.2 
.1     6.8 
8     6.4 

9.0   10.8 
8.5   10.2 
8.0     9.6 

12.6 
11.9 
11.2 

14.4 
13.6 

12.8 

16.2 
15.3 
14.4 

16 

COMMON  LOGARITHMS 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 

£80 

44716 

44731 

44747 

44762 

44778 

44793 

44809 

44824 

44840 

44855 

281 

44871 

44886 

44902 

44917 

44932 

44948 

44963 

44979 

44994 

45010 

282 

45025 

45040 

45056 

45071 

45086 

45102 

45117 

45133 

45148 

45163 

283 

45179 

45194 

45209 

45225 

45240 

45255 

45271 

45286 

45301 

45317 

284 

45332 

45347 

45362 

45378 

45393 

45408 

45423 

45439 

45454 

45469 

285 

45484 

45500 

45515 

45530 

45545 

45561 

45576 

45591 

45606 

45621 

286 

45637 

45652 

45667 

45682 

45697 

45712 

45728 

45743 

45758 

45773 

287 

45788 

45803 

45818 

45834 

45849 

45864 

45879 

45894 

45909 

45924 

288 

45939 

45954 

45969 

45984 

46000 

46015 

46030 

46045 

46060 

46075 

289 

46090 

46105 

46120 

46135 

46150 

46165 

461*80 

46195 

46210 

46225 

15 

290 

46240 

46255 

46270 

46285 

46300 

46315 

46330 

46345 

46359 

46374 

291 

46389 

46404 

46419 

46434 

46449 

46464 

46479 

46494 

46509 

46523 

292 

46538 

46553 

46568 

46583 

46598 

46613 

46627 

46642 

46657 

46672 

293 

46687 

46702 

46716 

46731 

46746 

46761 

46776 

46790 

46805 

46820 

294 

46835 

46850 

46864 

46879 

46894 

46909 

46923 

46938 

46953 

46967 

295 

46982 

46997 

47012 

47026 

47041 

47056 

47070 

47085 

47100 

47114 

296 

47129 

47144 

47159 

47173 

47188 

47202 

47217 

47232 

47246 

47261 

297 

47276 

47290 

47305 

47319 

47334 

47349 

47363 

47378 

47392 

47407 

298 

47422 

47436 

47451 

47465 

47480 

47494 

47509 

47524 

47538 

4755? 

299 

47567 

47582 

47596 

47611 

47625 

47640 

47654 

47669 

47683 

47698 

300 

47712 

47727 

47741 

47756 

47770 

47784 

47799 

47813 

4782S 

47842 

301 

47857 

47871 

47885 

47900 

47914 

47929 

47943 

47958 

47972 

47986 

302 

48001 

48015 

48029 

48044 

48058 

48073 

48087 

48101 

48116 

48130 

303 

48144 

48159 

48173 

48187 

48202 

48216 

48230 

48244 

48259 

48273 

304 

48287 

48302 

48316 

48330 

48344 

48359 

48373 

48387 

48401 

48416 

305 

48430 

48444 

48458 

48473 

48487 

48501 

48515 

48530 

48544 

48558 

306 

48572 

48586 

48601 

48615 

48629 

48643 

48657 

48671 

48686 

48700 

307 

48714 

48728 

48742 

48756 

48770 

48785 

48799 

48813 

48827 

48841 

308 

48855 

48869 

48883 

48897 

48911 

48926 

48940 

48954 

48968 

48982 

309 

48996 

49010 

49024 

49038 

49052 

49066 

49080 

49094 

49108 

49122 

14 

310 

49136 

49150 

49164 

49178 

49192 

49206 

49220 

49234 

49248 

49262 

311 

49276 

49290 

49304 

49318 

49332 

49346 

49360 

49374 

49388 

49402 

312 

49415 

49429 

49443 

49457 

49471 

49485 

49499 

49513 

49527 

49541 

313 

49554 

49568 

49582 

49596 

49610 

49624 

49638 

49651 

49665 

49679 

314 

49693 

49707 

49721 

49734 

49748 

49762 

49776 

49790 

49803 

49817 

315 

49831 

49845 

49859 

49872 

49886 

49900 

49914 

49927 

49941 

49955 

316 

49969 

49982 

49996 

50010 

50024 

50037 

50051 

50065 

50079 

50092 

317 

50106 

50120 

50133 

50147 

50161 

50174 

50188 

50202 

50215 

50229 

318 

50243 

50256 

50270 

50284 

50297 

50311 

50325 

50338 

50352 

50365 

319 

50379 

50393 

50406 

50420 

50433 

50447 

50461 

50474 

50488 

50501 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 

Proportional    Parts 


1.5 
1.4 


3.0 


4.5 
4.2 


6.0 
5.6 


7.5 

7.0 


9.0 

8.4 


10.5 
9.8 


12.0 
11.2 


13.5 
12.6 


COMMON    LOGARITHMS 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 

320 

321 

50515 
50651 

50529 
50664 

50542 

50678 

50556 
50691 

50569 
50705 

50583 
50718 

50596 
50732 

50610 
50745 

50623 
50759 

50637 

50772 

14 

322 
323 
324 
325 

50786 
50920 
51055 
51188 

50799 
50934 
51068 
51202 

50813 
50947 
51081 
51215 

50826 
50961 
51095 
51228 

50840 
50974 
51108 
51242 

50853 
50987 
51121 
51255 

50866 
51001 
51135 
51268 

50880 
51014 
51148 

51282 

50893 
51028 
51162 
51295 

50907 
51041 
51175 
51308 

326 
327 

328 

51322 
51455 
51587 

51335 

51468 
51601 

51348 
51481 
51614 

51362 
51495 
51627 

51375 

51508 
51640 

51388 
51521 
51654 

51402 
51534 
51667 

51415 
51548 
51680 

51428 
51561 
51693 

51441 
51574 
51706 

329 
330 

S31 

51720 
51851 
51983 

51733 
51865 
51996 

51746 
51878 
52009 

51759 
51891 
52022 

'51772 
51904 
52035 

51786 
51917 
52048 

51799 
51930 
52061 

51812 
51943 
52075 

51825 
51957 

52088 

51838 
51970 
52101 

332 
333 
334 
335 

336 
337 
338 

52114 
52244 
52375 
52504 

52634 
52763 
52892 

52127 
52257 
52388 
52517 

52647 
52776 
52905 

52140 
52270 
52401 
52530 

52660 
52789 
52917 

52153 
52284 
52414 
52543 

52673 
52802 
52930 

52166 
52297 
52427 
52556 

52686 
52815 
52943 

52179 
52310 
52440 
52569 

52699 
52827 
52956 

52192 
52323 
52453 
52582 

52711 
52840 
52969 

52205 
52336 
52466 
52595 

52724 
52853 
52982 

52218 
52349 
52479 
52608 

52737 

52866 
52994 

52231 
52362 
52492 
52621 

52750 
52879 
53007 

13 

339 
340 
341 

53020 
53148 
53275 

53033 
53161 
53288 

53046 
53173 
53301 

53058 
53186 
53314 

53071 
53199 
53326 

53084 
53212 
53339 

53097 
53224 
53352 

53110 
53237 
53364 

53122 
53250 
53377 

53135 
53263 
53390 

342 

343 
344 

53403 
53529 
53656 

53415 
53542 
53668 

'53428 
53555 
53681 

53441 
53567 
53694 

53453 
53580 
53706 

53466 
53593 
53719 

53479 
53605 
53732 

53491 
53618 
53744 

53504 
53631 
53757 

53517 
53648 
53769 

345 
346 
S47 

53782 
53908 
54033 

53794 
53920 
54045 

53807 
53933 
54058 

53820 
53945 
54070 

53832 
53958 
54083 

53845 
53970 
54095 

53857 
53983 
54108 

53870 
53995 
54120 

53882 
54008 
54133 

53895 
54020 
54145 

348 
349 

54158 
54283 

54170 
54295 

54183 
54307 

54195 
54320 

54208 
54332 

54220 
54345 

54233 
54357 

54245 

54370 

54258 
54382 

54270 
54394 

350 
351 
352 
353 

54407 
54531 
54654 
54777 

54419 
54543 
54667 
54790 

54432 
54555 

54679 
54802 

54444 
54568 
54691 
54814 

54456 
54580 
54704 
54827 

54469 
54593 
54716 
54839 

54481 
54605 
54728 
54851 

54494 
54617 
54741 
54864 

54506 
54630 
54753 
54876 

54518 
54642 
54765 
54888 

354 
355 
356 

54900 
55023 
55145 

54913 
55035 
55157 

54925 
55047 
55169 

54937 
55060 
55182 

54949 
55072 
55194 

54962 
55084 
55206 

54974 
55096 
55218 

54986 
55108 
55230 

54998 
55121 
55242 

55011 
55133 
55255 

12 

357 
358 
359 

55267 
55388 
55509 

55279 
55400 
55522 

55291 
55413 
55534 

55303 
55425 
55546 

55315 
55437 
55558 

55328 
55449 
55570 

55340 
55461 
55582 

55352 
55473 
55594 

55364 

55485 
55606 

55376 
55497 
55618 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Proportional    Parts 


D 

1 

2 

3 

4 

5 

6 

7 

8 

9 

14 
13 
12 

1.4 
1.3 
1.2 

2.8 
2.6 
2.4 

4.2 
3.9 
3.6 

5.6 
5.2 

4.8 

7.0 
6.5 
6.0 

8.4 
7.8 

7.2 

9.8 
9.1 

8.4 

11.2 
10.4 
9.6 

12.6 
11.7 
10.8 

CPMMON    LOGARITHMS 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 

360 
361 
362 
363 

55630 
55751 
55871 
55991 

55642 
55763 

55883 
56003 

55654 
55775 
55895 
56015 

55666 
55787 
55907 
56027 

55678 
55799 
55919 
56038 

55691 
55811 
55931 
56050 

55703 
55823 
55943 
56062 

55715 
55835 
55955 
56074 

55727 
55847 
55967 
56086 

55739 
55859 
55979 
56098 

364 
365 

366 

56110 
56229 
56348 

56122 
56241 
56360 

56134 
56253 
56372 

56146 
56265 
56384 

56158 
56277 
56396 

56170 
56289 
56407 

56182 
56301 
56419 

56194 
56312 
56431 

56205 
56324 
56443 

56217 
56336 
56455 

367 
368 
369 

56467 
56585 
56703 

56478 
56597 
56714 

56490 
56608 
56726 

56502 
56620 
56738 

56514 
56632 
56750 

56526 
56644 
56761 

56538 
56656 
56773 

56549 
56667 
56785 

56561 
56679 
56797 

56573 
56691 

56808 

12 

370 
371 
372 
373 

56820 
56937 
57054 
57171 

56832 
56949 
57066 
57183 

56844 
56961 
57078 
57194 

56855 
56972 
57089 
57206 

56867 
56984 
57101 
57217 

56879 
56996 
57113 
57229 

56891 
57008 
57124 
57241 

56902 
57019 
57136 
57252 

56914 
57031 
57148 
57264 

56926 
57043 
57159 
57276 

374 
375 
376 

.  57287 
57403 
57519 

57299 
57415 
57530 

57310 
57426 

57542 

57322 
57438 
57553 

57334 
57449 
57565 

57345 
57461 
57576 

57357 
57473 

57588 

57368 
57484 
57600 

57380 
57496 
57611 

57392 
57507 
57623 

377 

378 
379 

57634 
57749 
57864 

57646 
57761 

57875 

57657 

57772 
57887 

57669 
57784 
57898 

57680 
57795 
57910 

57692 
57807 
57921 

57703 
57818 
57933 

57715 
57830 
57944 

57726 
57841 
57955 

57738 
57852 
57967 

380 

381 
382 
383 

57978 
58092 
58206 
58320 

57990 
58104 
58218 
58331 

58001 
58115 
58229 
58343 

58013 
58127 
58240 
58354 

58024 
58138 
58252 
58365 

58035 
58149 
58263 

58377 

58047 
58161 

58274 
58388 

58058 
58172 
58286 
58399 

58070 
58184 
58297 
58410 

58081 
58195 
58309 
58422 

S84 
385 
386 

58433 
58546 
58659 

58444 
58557 
58670 

58456 
58569 
58681 

58467 
58580 
58692 

58478 
58591 
58704 

58490 
58602 
58715 

58501 
58614 
58726 

58512 
58625 
58737 

57524 
58636 
58749 

58535 
58647 
58760 

S87 
388 
389 

58771 
58883 
58995 

58782 
58894 
59006 

58794 
58906 
59017 

58805 
58917 
59028 

58816 
58928 
59040 

58827 
58939 
59051 

58838 
58950 
59062 

58850 
58961 
59073 

58861 
58973 
59084 

58872 
58984 
59095 

11 

390 
391 
392 
393 

59106 
59218 
59329 
59439 

59118 
59229 
59340 
59450 

59129 
59240 
59351 
59461 

59140 
59251 
59362 
59472 

59151 
59262 
59373 
59483 

59162 
59273 
59384 
59494 

59173 
59284 
59395 
59506 

59184 
59295 
59406 
59517 

59195 
59306 
59417 
59528 

59207 
59318 
59428 
59539 

394 
395 
396 

59550 
59660 
59770 

59561 
59671 
59780 

59572 
59682 
59791 

59583 
59693 
59802 

59594 
59704 
59813 

59605 
59715 
59824 

59616 
59726 
59835 

59627 
59737 
59846 

59638 
59748 
59857 

59649 
59759 
59868 

397 
398 
399 

59879 
59988 
60097 

59890 
59999 
60108 

59901 
60010 
60119 

59912 
60021 
60130 

59923 
60032 
60141 

59934 
60043 
60152 

59945 
60054 
60163 

59956 
60065 
60173 

59966 
60076 
60184 

59977 
60086 
60195 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 

Proportional    Parts 


1.2 
1.1 


2.4 

2.2 


3.6 
3.3 


4.8 
4.4 


6.0 
5.5 


7.2 
6.6 


8.4 

7.7 


9.6 

8.8 


10.8 
9.9 


COMMON    LOGARITHMS  19 


N0123456789D 

400  60206  60217  60228  60239  60249  60260  60271  60282  60293  60304 

401  60314  60325  60336  60347  60358  60369  60379  60390  60401  60412 

402  60423  60433  60444  60455  60466  60477  60487  60498  60509  60520 

403  60531  60541  60552  60563  60574  60584  60595  60606  60617  60627 

404  60638  60649  60660  60670  60681  60692  60703  60713  60724  60735 

405  60746  60756  60767  60778  60788  60799  60810  60821  60831  60842 

406  60853  60863  60874  60885  60895  60906  60917  60927  60938  60949  ^ 

407  60959  60970  60981  60991  61002  61013  61023  61034  61045  61055 

408  61066  61077  61087  61098  61109  61119  61130  61140  61151  61162 

409  61172  61183  61194  61204  61215  61225  61236  61247  61257  61268 

410  61278  61289  61300  61310  61321  61331  61342  61352  61363  61374 

411  61384  61395  61405  6141G  61426  61437  61448  61458  61469  61479 

412  61490  61500  61511  61521  61532  61542  61553  61563  61574  61584 

413  61595  61606  61616  61627  61637  61648  61658  61669  61679  61690 

414  61700  61711  61721  61731  61742  61752  61763  61773  61784  61794 

415  61805  61815  61826  61836  61847  61857  61868  61878  61888  61899 

416  61909  61920  61930  61941  61951  61962  61972  61982  61993  62003 

417  62014  62024  62034  62045  62055  H2066  62076  62086  62097  62107 

418  62118  62128  62138  62149  62159  62170  62180  62190  62201  62211 

419  62221  62232  62242  62252  62263  62273  62284  62294  62304  62315 

420  62325  62335  62346  62356  62366  62377  62387  62397  62408  62418 

421  62428  62439  62449  62459  62469  62480  62490  62500  62511  62521 

422  62531  62542  62552  62562  62572  62-583  62593  62603  62613  62624 

423  62634  62644  62655  62665  62675  62685  62696  62706  62716  62726 

424  62737  62747  62757  62767  62778  62788  62798  62808  62818  62829 

425  62839  62849  62859  62870  62880  62890  62900  62910  62921  62931 

426  62941  62951  62961  62972  62982  62992  63002  63012  63022  63033 

427  63043  63053  63063  63073  63083  63094  63104  63114  63124  63134 

428  63144  63155  63165  63175  63185  63195  63205  63215  63225  63236 

429  63246  63256  63266  63276  63286  63296  63306  63317  63327  63337 

430  63347  63357  63367  63377  63387  63397  63407  63417  63428  63438 

431  63448  63458  63468  63478  63488  63498  63508  63518  63528  63538  10 

432  63548  63558  63568  63579  63589  63599  63609  63619  63629  63639 

433  63649  63659  63669  63679  63689  63699  63709  63719  63729  63739 

434  63749  63759  63769  63779  63789  63799  63809  63819  63829  63839 

435  63849  63859  63869  63879  63889  63899  "63909  63919  63929  63939 

436  63949  63959  63969  63979  63988  63998  64008  64018  64028  64038 

437  64048  64058  64068  64078  64088  64098  64108  64118  64128  64137 

438  64147  64157  64167  64177  64187  64197  64207  64217  64227  64237 

439  64246  64256  64266  «4276  64286  64296  64306  64316  64326  64335 

440  64345  64355  64365  64375  64385  64395  64404  64414  64424  64434 

441  64444  64454  64464  64473  64483  64493  64503  64513  64523  64532 

442  64542  64552  64562  64572  64582  64591  64601  64611  64621  64631 

443  64640  64650  64660  64670  64680  64689  64699  64709  64719  64729 

444  64738  64748  64758  64768  64777  64787  64797  64807  64816  64826 

445  64836  64846  64856  64865  64875  64885  64895  64904  64914  64924 

446  64933  64943  64953  64963  64972  64982  64992  65002  65011  65021 

447  65031  65040  65050  65060  65070  65079  65089  65099  65108  65118 

448  65128  65137  65147  65157  65167  65176  65186  65196  65205  65215 

449  65225  65234  65244  65254  65263  65273  65283  65292  65302  65312 

N  01     23     4     56  7     8     9    D 


COMMON    LOGARITHMS 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 

450 
451 
452 
453 

65321 
65418 
65514 
65610 

65331 
65427 
65523 
65619 

65341 
65437 
65533 
65629 

65350 
65447 
65543 
65639 

65360 
65456 
65552 
6564S 

65369 
65466 
65562 
65658 

65379 
65475 
65571 
65667 

65389 
65485 
65581 
65677 

65398 
65495 
65591 
65686 

65408 
65504 
65600 
65696 

10 

454 
455 
456 

65706 
65801 
65896 

65715 
65811 
65906 

65725 

65820 
65916 

65734 

65830 
65925 

65744 
65839 
65935 

65753 
65849 
65944 

65763 
65858 
65954 

65772 
65868 
65963 

65782 
65877 
65973 

65792 

65887 
65982 

457 

458 
459 

65992 
66087 
66181 

66001 
66096 
66191 

66011 
66106 
66200 

66020 
66115 
66210 

66030 
66124 
66219 

66039 
66134 
66229 

66049 
66143 
66238 

66058 
66153 
66247 

66068 
66162 
66257 

66077 
66172 
66266 

460 
461 
462 
463 

66276 
66370 
66464 
66558 

66285 
66380 
66474 
66567 

66295 
66389 
66483 
66577 

66304 
66398 
66492 
66586 

66314 

66408 
66502 
66596 

66323 
66417 
66511 
66605 

66332 
66427 
66521 
66614 

66342 
66436 
66530 
66624 

66351 
66445 
66539 
66633 

66361 
66455 
66549 
66642 

464 
465 
466 

66652 
66745 
66839 

66661 
66755 
66848 

66671 
66764 
66857 

66680 
66773 
66867 

66689 
66783 
66876 

66699 
66792 
66885 

66708 
66801 
66894 

66717 
66811 
66904 

66727 
66820 
66913 

66736 
66829 
66922 

467 

468 
469 

66932 
67025 
67117 

66941 
67034 
67127 

66950 
67043 
67136 

66960 
67052 
67145 

66969 
67062 
67154 

66978 
67071 
67164 

66987 
67080 
67173 

66997 

67089 
67182 

67006 
67099 
67191 

67015 
67108 
67201 

470 
471 
472 
473 

67210 
67302 
67394 
67486 

67219 
67311 
67403 
67495 

67228 
67321 
67413 
67504 

67237 
67330 
67422 
67514 

67247 
67339 
67431 
67523 

67256 
67348 
67440 
67532 

67265 
67357 
67449 
67541 

67274 
67367 
67459 
67550 

67284 
67376 
67468 
67560 

67293 
67385 
67477 
67569 

474 
475 
476 

477 
478 
479 

67578 
67669 
67761 

67852 
67943 
68034 

67587 
67679 
67770 

67861 
67952 
68043 

67596 
67688 
67779 

67870 
67961 
68052 

67605 
67697 
67788 

67879 
67970 
68061 

67614 
67706 
67797 

67888 
67979 
68070 

67624 
67715 
67806 

67897 
67988 
68079 

67633 
67724 
67815 

67906 
67997 
68088 

67642 
67733 

67825 

67916 
68006 
68097 

67651 
67742 
67834 

67925 
68015 
68106 

67660 
67752 
67843 

67934 

68024 
68115 

9 

480 

481 
482 
483 

68124 
68215 
68305 
68395 

68133 
68224 
68314 
68404 

68142 
68233 
68323 
68413 

68151 
68242 
68332 
68422 

68160 
68251 
68341 
68431 

68169 
68260 
68350 
68440 

68178 
68269 
68359 
68449 

68187 
68278 
68368 
68458 

68196 
68287 
68377 
68467 

68205 
68296 
68386 
68476 

184 
485 
486 

68485 
68574 
68664 

68494 
68583 
68673 

68502 
68592 
68681 

68511 
68601 
68690 

68520 
68610 
68699 

68529 
68619 

68708 

68538 
68628 
68717 

68547 
68637 
68726 

68556 
68646 
68735 

68565 
68655 

68744 

487 
488 
489 

68753 
68842 
68931 

68762 
68851 
68940 

68771 
68860 
68949 

68780 
68869 
68958 

68789 
68878 
68966 

68797 
68886 
68975 

68806 
68895 
68984 

68815 
68904 
68993 

68824 
68913 
69002 

68833 
68922 
69011 

490 
491 
492 
493 

69020 
69108 
69197 
69285 

69028 
69117 
69205 
69294 

69037 
69126 
69214 
69302 

69046 
69135 
69223 
69311 

69055 
69144 
69232 
69320 

69064 
69152 
69241 
69329 

69073 
69161 
69249 
69338 

69082 
69170 
69258 
69346 

69090 
69179 
69267 
69355 

69099 
69188 
69276 
69364 

494 
495 
496 

69373 
69461 
69548. 

69381 
69469 
69557 

69390 
69478 
69566 

69399 
69487 
69574 

69408 
69496 
69583 

69417 
69504 
69592 

69425 
69513 
69601 

69434 
69522 
69609 

69443 
69531 
69618 

69452 
69539 
69627 

497 
498 
499 

69636 
69723 
69810 

69644 
69732 
69819 

69653 
69740 

69827 

69662 
69749 
69836 

69671 
69758 
69845 

69679 
69767 
69854 

69688 
69775 
69862 

69697 
69784 
69871 

69705 
69793 
69880 

69714 
69801 
69888 

COMMON  LOGARITHMS 

21 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9    D 

500 

501 
502 
503 

69897 
69984 
70070 
70157 

69906 
69992 
70079 
70165 

69914 
70001 
70088 
70174 

69923 
70010 
70096 
70183 

69932 
70018 
70105 
70191 

69940 
70027 
70114 
70200 

69949 
70036 
70122 
70209 

69958 
70044 
70131 
70217 

69966 
70053 
70140 
70226 

69975 
70062 
70148 
70234 

504 
505 
506 

70243 
70329 
70415 

70252 
70338 
70424 

70260 
70346 
70432 

70269 
70355 
70441 

70278 
70364 
70449 

70286 
70372 
70458 

70295 
70381 
70467 

70303 
70389 
70475 

70312 
70398 

70484 

70321 
70406 
70492  9 

507 
508 
509 

70501 
70586 
70672 

70509 
70595 
70680 

70518 
70603 
70689 

70526 
70612 
70697 

70535 
70621 
70706 

70544 
70629 
70714 

70552 
70638 
70723 

70561 
70646 
70731 

70569 
70655 
70740 

70578 
70663 
70749 

510 
511 
512 

70757 
70842 
70927 

70766 
70851 
70935 

70774 
70859 
70944 

70783 
70868 
70952 

70791 
70876 
70961 

70800 
70885 
70969 

70808 
70893 
70978 

70817 
70902 
70986 

70825 
70910 
70995 

70834 
70919 
71003 

513 
514 
515 
516 

71012 
71096 
71181 
71265 

71020 
71105 
71189 

71273 

71029 
71113 

71198 
71282 

71037 
71122 
71206 
71290 

71046 
71130 
71214 
71299 

71054 
71139 
71223 
71307 

71063 
71147 
71231 
71315 

71071 
71155 
71240 
71324 

71079 
71164 
71248 
71332 

71088 
71172 
71257 
71341 

517 
518 
519 

71349 
71433 
71517 

71357 
71441 
71525 

71366 
71450 
71533 

71374 
71458 
71542 

71383 
71466 
71550 

71391 
71475 
71559 

71399 
71483 
71567 

71408 
71492 
71575 

71416 
71500 
71584 

71425 
71508 
71592 

520 

521 
522 
523 

71600 
71684 
71767 

71850 

71609 
71692 
71775 
71858 

71617 
71700 
71784 
71867 

71625 
71709 
71792 

71875 

71634 
71717 
71800 
71883 

71642 
71725 
71809 
71892 

71650 
71734 
71817 
71900 

71659 
71742 
71825 
71908 

71667 
71750 
71834 
71917 

71675 
71759 
71842 
71925 

524 
525 

526 

71933 
72016 
72099 

71941 
72024 
72107 

71950 
72032 
72115 

71958 
72041 
72123 

71966 
72049 
72132 

71975 

72057 
72140 

71983 
72066 
72148 

71991 
72074 
72156 

71999 
72082 
72165 

72008 
72090 
72173 

527 
528 
529 

72181 
72263 
72346 

72189 

72272 
72354 

72198 
72280 
72362 

72206 
72288 
72370 

72214 
72296 
72378 

72222 
72304 
72387 

72230 
72313 
72395 

7^239 
72321 
72403 

72247 
72329 
72411 

72255 
72337 
72419 

530 
531 
532 
533 

72428 
72509 
72591 
72673 

72436 
72518 
72599 
72681 

72444 
72526 
72607 
72689 

72452 
72534 
72616 
72697 

72460 
72542 
72624 
72705 

72469 
72550 
72632 
72713 

72477 
72558 
72640 
72722 

72485 
72567 
72648 
72730 

72493 

72575 
72656 
.72738 

72501   a 
72583 
72665 
72746 

534 
535 

536 

72754 
72835 
72916 

72762 
72843 

72925 

72770 
72852 
72933 

72779 
72860 
72941 

72787 
72868 
72949 

72795 

72876 
72957 

72803 
72884 
72965 

72811 

72892 
72973 

72819 
72900 
72981 

72827 
72908 
72989 

537 
538 
539 

72997 
73078 
73159 

73006 
73086 
73167 

73014 
73094 
73175 

73022 
73102 
73183 

73030 
73111 
73191 

73038 
73119 
73199 

73046 
73127 
73207 

73054 
73135 
73215 

73062 
73143 
73223 

73070 
73151 
73231 

540 
541 
542 
543 

73239 
73320 
73400 
73480 

73247 
73328 
73408 
73488 

73255 
73336 
73416 
73496 

73263 
73344 
73424 
73504 

73272 
73352 
73432 
73512 

73280 
73360 
73440 
73520 

73288 
73368 
73448 
73528 

73296 
73376 
73456 
73536 

73304 
73384 
73464 
73544 

73312 
73392 
73472 
73552 

544 
545 
546 

73560 
73640 
73719 

73568 
73648 
73727 

73576 
73656 
73735 

73584 
73664 
73743 

73592 
73672 
73751 

73600 
73679 
73759 

73608 
73687 
73767 

73616 
73695 
73775 

73621 
73703 
73783 

73632 
73711 
73791 

547 

548 
549 

73799 
73878 
73957 

73807 
73886 
73965 

73815 
73894 
73973 

73823 
73902 
73981 

73830 
73910 
73989 

73838 
73918 
73997 

73846 
73926 
74005 

73854 
73933 
74013 

73862 
73941 

74020 

73870 
73949 
74028 

COMMON    LOGARITHMS 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 

550 

551 
552 
553 

74036 
74115 
74194 
74273 

74044 
74123 
74202 

74280 

74052 
74131 
74210 

74288 

74060 
74139 
74218 
74296 

74068 
74147 
74225 
74304 

74076 
74155 
74233 
74312 

74084 
74162 
74241 
74320 

74092 
74170 
74249 
74327 

74099 
74178 
74257 
74335 

74107 
74186 
74265 
74343 

554 
555 
556 

74351 

74429 
74507 

74359 
74437 
74515 

74367 
74445 
74523 

74374 
74453 
74531 

74382 
74461 
74539 

74390 
74468 
74547 

74398 
74476 
74554 

74406 
74484 
74562 

74414 
74492 
74570 

74421 
74500 

74578 

557 
558 
559 

74586 
74663 
74741 

74593 
74671 
74749 

74601 
74679 

74757 

74609 
74687 
74764 

74617 
74695 

74772 

74624 
74702 
74780 

74632 
74710 

74788 

74640 

74718 
74796 

74648 
74726 
74803 

74656 
74733 
74811 

560 
561 
562 
563 

74819 
74896 
74974 
75051 

74827 
74904 
74981 
75059 

74834 
74912 
74989 
75066 

74842 
74920 
74997 
75074 

74850 
74927 
75005 
75082 

74858 
74935 
75012 
75089 

74865 
74943 
75020 

75097 

74873 
74950 
75028 
75105 

74881 
74958 
75035 
75113 

74889 
74966 
75043 
75120 

564 
565 
566 

75128 
75205 
75282 

75136 
75213 
75289 

75143 

75220 
75297 

75151 
75228 
75305 

75159 
75236 
75312 

75166 
75243 
75320 

75174 
75251 
75328 

75182 
75259 
75335 

75189 
75266 
75343 

75197 
75274 
75351 

8 

567 
568 
569 

75358 
75435 
75511 

75366 

75442 
75519 

75374 
75450 
75526 

75381 
75458 
75534 

75389 
75465 
75542 

75397 
75473 
75549 

75404 
75481 
75557 

75412 
75488 
75565 

75420 
75496 

75572 

75427 
75504 
75580 

570 
571 
572 
573 

75587 
75664 
75740 
75815 

75595 
75671 

75747 
75823 

75603 
75679 
75755 
75831 

75610 
75686 
75762 
75838 

75618 
75694 
75770 
75846 

75626 
75702 

75778 
75858 

75633 
75709 

75785 
75861 

75641 
75717 
75793 
75868 

75648 
75724 
75800 
75876 

75656 
75732 

75808 
75884 

574 
575 
576 

75891 
75967 
76042 

75899 
75974 
76050 

75906 

75982 
76057 

75914 
75989 
76065 

75921 

75997 
76072 

75929 
76005 
76080 

75937 
76012 

76087 

75944 
76020 
76095 

75952 
76027 
76103 

75959 
76035 
76110 

577 
578 
579 
580 

581 
582 
583 

76118 
76193 
76268 
76343 

76125 
76200 
76275 
76350 

76133 
76208 
76283 
76358 

76140 
76215 
76290 
76365 

76148 
76223 
76298 
76373 

76155 
76230 
76305 
76380 

76163 

76238 
76313 
76388 

76170 
76245 
76320 
76395 

76178 
76253 
76328 
76403 

76185 
76260 
76335 
76410 

76418 
76492 
76567 

76425 
76500 
76574 

76433 

76507 
76582 

76440 
76515 
76589 

76448 
76522 
76597 

76455 
76530 
76604 

76462 
76537 
76612 

76470 
76545 
76619 

76477 
76552 
76626 

76485 
76559 
76634 

584 
585 

586 

76641 
76716 
76790 

76649 
76723 
76797 

76656 
76730 
76805 

76664 
76738 
76812 

76671 
76745 
76819 

76678 
76753 

76827 

76686 
76760 
76834 

76693 

76768 
76842 

76701 

76775 
76849 

76708 
76782 
76856 

587 
588 
589 

76864 
76938 
77012 

76871 
76945 
77019 

76879 
76953 
77026 

76886 
76960 
77034 

76893 
76967 
77041 

76901 
76975 
77048 

76908 
76982 
77056 

76916 
76989 
77063 

76923 
76997 
77070 

76930 
77004 
77078 

590 
591 
592 
593 

77085 
77159 
77232 
77305 

77093 
77166 
77240 
77313 

77100 
77173 
77247 
77320 

77107 
77181 
77254 
77327 

77115 
77188 
77262 
77335 

77122 
77195 
77269 
77342 

77129 
77203 
77276 
77349 

77137 
77210 
77283 
77357 

77144 
77217 
77291 
77364 

77151 

77225 
77298 
77371 

7 

594 
595 

596 

77379 
77452 
77525 

77386 
77459 
77532 

77393 
77466 
77539 

77401 

77474 
77546 

77408 
77481 
77554 

77415 
77488 
77561 

77422 
77495 
77568 

77430 
77503 
77576 

77437 
77510 
77583 

77444 
77517 
77590 

597 

598 
599 

77597 
77670 
77743 

77605 
77677 
77750 

77612 
77685 

77757 

77619 
77692 
77764 

77627 
77699 

77772 

77634 
77706 
77779 

77641 
77714 

77786 

77648 
77721 
77793 

77656 

77728 
77801 

77663 
77735 

77808 

COMMON    LOGARITHMS  23 


N0123456789 

600  77815.  77822  77830  77837  77844  77851  77859  77866  77873  77880 

601  77887  77895  77902  77909  77916  77924  77931  77938  77945  77952 

602  77960  77967  77974  77981  77988  77996  78003  78010  78017  78025 

603  78032  78039  78046  78053  78061  78068  78075  78082  78089  78097 

604  78104  78111  78118  78125  78132  78140  78147  78154  78161  78168 

605  78176  78183  78190  78197  78204  78211  78219  78226  78233  78240 

606  78247  78254  78262  78269  78276  78283  78290  78297  78305  78312 

607  78319  78326  78333  78340  78347  78355  78362  78369  78376  78383 

608  78390  78398  78405  78412  78419  78426  78433  78440  78447  78455 

609  78462  78469  78476  78483  78490  78497  78504  78512  78519  78526 

610  78533  78540  78547  78554  78561  78569  78576  78583  78590  78597 

611  78604  78611  78618  78625  78633  78640  78647  78654  78661  78668 

612  78675  78682  78689  78696  78704  78711  78718  78725  78732  78739 

613  78746  78753  78760  78767  78774  78781  78789  78796  78803  78810 

614  78817  78824  78831  78838  78845  78852  78859  78866  78873  78880 

615  78888  78895  78902  78909  78916  78923  78930  78937  78944  78951 

616  78958  78965  78972  78979  78986  78993  79000  79007  79014  79021 

617  79029  79036  79043  79050  79057  79064  79071  79078  79085  79092 

618  79099  79106  79113  79120  79127  79134  79141  79148  79155  79162 

619  79169  79176  79183  79190  79197  79204  79211  79218  79225  79232 

620  79239  79246  79253  79260  79267  79274  79281  79288  79295  79302 

621  79309  79316  79323  79330  79337  79344  79351  79358  79365  79372 

622  79379  79386  79393  79400  79407  79414  79421  79428  79435  79442 

623  79449  79456  79463  79470  79477  79484  79491  79498  79505  79511 

624  79518  79525  79532  79539  79546  79553  79560  79567  79574  79581 

625  79588  79595  79602  79609  79616  79623  79630  79637  79644  79650 

626  79657  79664  79671  79678  79685  79692  79699  79706  79713  79720 

627  79727  79734  79741  79748  79754  79761  79768  79775  79782  79789 

628  79796  79803  79810  79817  79824  79831  79837  79844  79851  79858 

629  79865  79872  79879  79886  79893  79900  79906  79913  79920  79927 

630  79934  79941  79948  79955  79962  79969  79975  79982  79989  79996 

631  80003  80010  80017  80024  80030  80037  80044  80051  80058  80065 

632  80072  80079  80085  80092  80099  80106  80113  80120  80127  80134 

633  .  80140  80147  80154  80161  80168  80175  80182  80188  80195  80202 

634  80209  80216  80223  80229  80236  80243  80250  80257  80264  80271 

635  80277  80284  80291  80298  80305  80312  80318  80325  80332  80339 

636  80346  80353  80359  80366  80373  80380  80387  80393  80400  80407 

637  80414  80421  80428  80434  80441  80448  80455  80462  80468  80475 

638  80482  80489  80496  80502  80509  80516  80523  80530  80536  80543 

639  80550  80557  80564  80570  80577  80584  80591  80598  80604  80611 

640  80618  80625  80632  80638  80645  80652  80659  80665  80672  80679 

641  80686  80693  80699  80706  80713  80720  80726  80733  80740  80747 

642  80754  80760  80767  80774  80781  80787  80794  80801  80808  80814 

643  80821  80828  80835  80841  80848  80855  80862  80868  80875  80882 

644  80889  80895  80902  80909  80916  80922  80929  80936  80943  80949 

645  80956  80963  80969  80976  80983  80990  80996  81003  81010  81017 

646  81023  81030  81037  81043  81050  81057  81064  81070  81077  81084 

647  81090  81097  81104  81111  81117  81124  81131  81137  81144  81151 

648  81158  81164  81171  81178  81184  81191  81198  81204  81211  81218 

649  81224  81231  81238  81245  81251  81258  81265  81271  81278  81285 

N0123456789 


24 

COMMON  LOGARITHMS 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 

650 

651 
652 
653 

81291 
81358 
81425 
81491 

81298 
81365 
81431 
81498 

81305 
81371 
81438 
81505 

81311 

81378 
81445 
81511 

81318 
81385 
81451 
81518 

81325 
81391 
81458 

81525 

81331 

81398 
81465 
81531 

81338 
81405 
81471 
81538 

81345 
81411 
81478 
81544 

81351 

81418 
81485 
81551 

654 
655 

656 

81558 
81624 
81690 

81564 
81631 
81697 

81571 
81637 
81704 

81578 
81644 
81710 

81584 
81651 
81717 

81591 
81657 
81723 

81598 
81664 
81730 

81604 
81671 
81737 

81611 
81677 
81743 

81617 
81684 
81750 

657 

658 
659 

81757 
81823 
81889 

81763 
81829 
81895 

81770 
81836 
81902 

81776 
81842 
81908 

81783 
81849 
81915 

81790 
81856 
81921 

81796 
81862 
81928 

81803 
81869 
81935 

81809 
81875 
81941 

81816 
81882 
81948 

660 
661 
662 
663 

81954 
82020 
82086 
82151 

81961 
82027 
82092 
82158 

81968 
82033 
82099 
82164 

81974 
82040 
82105 
82171 

81981 
82046 
82112 
82178 

81987 
82053 
82119 
82184 

81994 
82060 
82125 
82191 

82000 
82066 
82132 
82197 

82007 
82073 
82138 
82204 

82014 
82079 
82145 

82210 

7 

664 
665 

666 

82217 
82282 
82347 

82223 
82289 
82354 

82230 
82295 
82360 

82236 
82302 
82367 

82243 
82308 
82373 

82249 
82315 
82380 

82256 
82321 
82387 

82263 
82328 
82393 

82269 
82334 

82400 

82276 
82341 
82406 

667 
668 
669 
670 

82413 
82478 
82543 
82607 

82419 
82484 
82549 
82614 

82426 
82491 
82556 
82620 

82432 
82497 
82562 
82627 

82439 
82504 
82569 
82633 

82445 
82510 
82575 
82640 

82452 
82517 
82582 
82646 

82458 
82523 
82588 
82653 

82465 
82530 
82595 
82659 

82471 
82536 
82601 
82666 

671 
672 
673 

82672 

82737 
82802 

82679 
82743 

82808 

82685 
82750 
82814 

82692 
82756 
82821 

82698 
82763 

82827 

82705 

82769 
82834 

82711 

82776 
82840 

82718 

82782 
82847 

82724 
82789 
82853 

82730 

82795 
82860 

674 
F75 

676 

82866 
82930 
82995 

82872 
82937 
83001 

82879 
82943 
83008 

82885 
82950 
83014 

82892 
82956 
83020 

82898 
82963 
83027 

82905 
82969 
83033 

82911 
82975 
83040 

82918 
82982 
83046 

82924 

82988 
83052 

677 
678 
679 

83059 
83123 

83187 

83065 
83129 
83193 

83072 
83136 
83200 

83078 
83142 
83206 

83085 
83149 
83213 

83091 
83155 
83219 

83097 
83161 
83225 

83104 
83168 
83232 

83110 
83174 
83238 

83117 
83181 
83245 

680 

681 
682 
683 

83251 
83315 
83378 
83442 

83257 
83321 
83385 
83448 

83264 
83327 
83391 
83455 

83270 
83334 
83398 
83461 

83276 
83340 
83404 
83467 

83283 
83347 
83410 
83474 

83289 
83353 
83417 

83480 

83296 
83359 
83423 
83487 

83302 
833G6 
83429 
83493 

83308 
83372 
83436 
83499 

684 
685 

686 

83506 
83569 
83632 

83512 
83575 
83639 

83518 
83582 
83645 

83525 
8358S 
83651 

83531 
83594- 
83658 

83537 
83601 
83664 

83544 
83607 
83670 

83550 
83613 
83677 

83556 
83620 
83683 

83563 
83626 
83689 

6 

687 
688 
689 

83696 
83759 
83822 

83702 
83765 

83828 

83708 
83771 
83835 

83715 

83778 
83841 

83721 
83784 
83847 

83727 
83790 
83853 

83734 
83797 
83860 

83740 
83803 
83866 

83746 
83809 
83872 

83753 
83816 
83879 

690 

691 
692 
693 

83885 
83948 
84011 
84073 

83891 
83954 
84017 
84080 

83897 
83960 
84023 
84086 

83904 
83967 
84029 
84092 

83910 
83973 
84036 

84098 

83916 
83979 
84042 
84105 

83923 
83985 
84048 
84111 

83929 
83992 
84055 
84117 

83935 
83998 
84061 
84123 

83942 
84004 
84067 
84130 

694 
695 

696 

84136 
84198 
84261 

84142 
84205 
84267 

84148 
84211 
84273 

84155 
84217 

84280 

84161 
84223 

84286 

84167 
84230 
84292 

84173 
84236 

84298 

84180 
84242 
84305 

84186 
84248 
84311 

84192 
84255 
84317 

697 
698 
699 

84323 
84386 
84448 

84330 
84392 
84454 

84336 
84398 
84460 

84342 
84404 
84466 

84348 
84410 
84473 

84354 
84417 
84479 

84361 
84423 

84485 

84367 
84429 
84491 

84373 
84435 
84497 

84379 
84442 
84504 

COMMON  LOGARITHMS 

* 

25 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9    D 

700 

701 
702 
703 

84510 
84572 
84634 
84696 

84516 

84578 
84640 
84702 

84522 
84584 
84646 

84708 

84528 
84590 
84652 
84714 

84535 
84597 
84658 
84720 

84541 
84603 
84665 
84726 

84547 
84609 
84671 
84733 

84553 
84615 
84677 
84739 

84559 
84621 
84683 
84745 

84566 
84628 
84689 
84751 

704 
705 
706 

84757 
84819 
84880 

84763 

84825 

84887 

84770 
84831 
84893 

84776 
84837 
84899 

84782 
84844 
84905 

84788 
84850 
84911 

84794 
84856 
84917 

84800 
84862 
84924 

84807 
84868 
84930 

84813 
84874 
84936 

707 
708 
709 

84942 
85003 

85065 

84948 
85009 
85071 

84954 
85016 

85077 

84960 
85022 
85083 

84967 
85028 
85089 

84973 
85034 
85095 

84979 
85040 
85101 

84985 
85046 
85107 

84991 
85052 
85114 

84997 
85058 
85120 

710 

711 
712 
713 

85126 
85187 
85248 
85309 

85132 
85193 
85254 
85315 

85138 
85199 
85260 
85321 

85144 
85205 
85266 
85327 

85150 
85211 
85272 
85333 

85156 
85217 
85278 
85339 

85163 
85224 
85285 
85345 

85169 
85230 
85291 
85352 

85175 
85236 
85297 
85358 

85181 
85242 
85303 
85364 

714 
715 

716 

85370 
85431 
85491 

85376 
85437 
85497 

85382 
85443 
85503 

85388 
85449 
85509 

85394 
85455 
85516 

85400 
85461 
85522 

85406 
85467 

85528 

85412 

85473 
85534 

85418 
85479 
85540 

85425 

85485 
85546 

717 

718 
719 

85552 
85612 
85673 

85558 
85618 
85679 

85564 
85625 

85685 

85570 
85631 
85691 

85576 
85637 
85697 

85582 
85643 
85703 

85588 
85649 
85709 

85594 
85655 
85715 

85600 
85661 
85721 

85606 
85667 

85727 

720 
721 
722 
723 

85733 

85794 
85854 
85914 

85739 

85800 
85860 
85920 

85745 
85806 
85866 
85926 

85751 
85812 
85872 
85932 

85757 
85818 
85878 
85938 

85763 

85824 
85884 
85944 

85769 
85830 
85890 
85950 

85775 
85836 
85896 
85956 

85781 
85842 
85902 
85962 

85788 
85848 
85908 
85968 

724 
725 

726 

85974 
86034 
86094 

85980 
86040 
86100 

85986 
86046 
86106 

85992 
86052 
86112 

85998 
86058 
86118 

86004 
86064 
86124 

86010 
86070 
86130 

86016 
86076 
86136 

86022 
86082 
86141 

86028   . 
86088   * 
86147 

727 
728 
729 

86153 
86213 
86273 

86159 
86219 
86279 

86165 
86225 

86285 

86171 
86231 
86291 

86177 
86237 
86297 

86183 
86243 
86303 

86189 
86249 
86308 

86195 
86255 
86314 

86201 
86261 
86320 

86207 
86267 
86326 

730 
731 
732 
733 

86332 
86392 
86451 
86510 

86338 
86398 
86457 
86516 

86344 
86404 
86463 
86522 

86350 
86410 
86469 
86528 

86356 
86415 
86475 
86534 

86362 
86421 
86481 
86540 

86368 
86427 
86487 
86546 

86374 
86433 
86493 
86552 

86380 
86439 
86499 
86558 

86386 
86445 
86504 
86564 

734 
735 
736 

86570 
86629 
86688 

86576 
86635 
86694 

86581 
86641 
86700 

86587 
86646 
86705 

86593 
86652 
86711 

86599 
86658 

86717 

86605 
86664 
86723 

86611 
86670 
86729 

86617 
86676 
86735 

86623 
86682 
86741 

737 
738 
739 

86747 
86806 
86864 

86753 
86812 

86870 

86759 

86817 
86876 

86764 
86823 
86882 

86770 
86829 
86888 

86776 
86835 
86894 

86782 
86841 
86900 

86788 
86847 
86906 

86794 
86853 
86911 

86800 
86859 
86917 

740 
741 
742 
743 

86923 
86982 
87040 
87099 

86929 
86988 
87046 
87105 

86935 
86994 
87052 
87111 

86941 
86999 
87058 
87116 

86947 
87005 
87064 
87122 

86953 
87011 

87070 
87128 

86958 
87017 
87075 
87134 

86964 
87023 
87081 
87140 

86970 
87029 
87087 
87146 

86976 
87035 
87093 
87151 

744 
745 
746 

87157 
87216 

87274 

87163 
87221 

87280 

87169 

87227 
87286 

87175 
87233 
87291 

87181 
87239 

87297 

87186 
87245 
87303 

87192 
87251 
87309 

87198 
87256 
87315 

87204 
87262 
87320 

87210 
87268 
87326 

747 
748 
749 

87332 
87390 

87448 

87338 
87396 
87454 

87344 
87402 
87460 

87349 

87408 
87466 

87355 
87413 
87471 

87361 
87419 

87477 

87367 
87425 
87483 

87373 
87431 

87489 

87379 
87437 
87495 

87384 
87442 
87500 

26 


COMMON    LOGARITHMS 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9    D 

750 

751 
752 
753 

87506 
87564 
87622 
87679 

87512 

87570 
87628 
87685 

87518 
87576 
87633 
87691 

87523 
87581 
87639 
87697 

87529 
87587 
87645 
87703 

87535 
87593 
87651 
87708 

87541 
87599 
87656 
87714 

87547 
87604 
87662 
87720 

87552 
87610 
87668 
87726 

87558 
87616 
87674 
87731 

754 
755 

756 

87737 
87795 
87852 

87743 

87800 
87858 

87749 
87806 
87864 

87754 
87812 
87869 

87760 
87818 

87875 

87766 
87823 
87881 

87772 
87829 
87887 

87777 
87835 
87892 

87783 
87841 
87898 

87789 
87846 
87904 

757 
758 
759 

87910 
87967 
88024 

87915 
87973 
88030 

87921 
87978 
88036 

87927 
87984 
88041 

87933 
87990 
88047 

87938 
87996 
88053 

87944 
88001 
88058 

87950 
88007 
88064 

87955 
88013 
88070 

87961 
88018 
88076 

760 

761 
762 
763 

88081 
88138 
88195 
88252 

88087 
88144 
88201 

88258 

88093 
88150 
88207 
88264 

88098 
88156 
88213 

88270 

88104 
88161 
88218 
88275 

88110 
88167 
88224 
88281 

88116 
88173 
88230 

88287 

88121 
88178 
88235 

88292 

88127 
88184 
88241 
88298 

88133 
88190 
88247 
88304 

764 
765 

766 

88309 
88366 
88423 

88315 
88372 
88429 

88321 
88377 
88434 

88326 
88383 
88440 

88332 
88389 
88446 

88338 
88395 
88451 

88343 
88400 
88457 

88349 
88406 
88463 

88355 

88412 
88468 

88360 
88417 
88474 

767 
768 
769 

88480 
88536 
88593 

88485 
88542 
88598 

88491 
88547 
88604 

88497 
88553 
88610 

88502 
88559 
88615 

88508 
88564 
88621 

88513 
88570 
88627 

88519 
88576 
88632 

88525 
88581 
88638 

88530 
88587 
88643 

770 

771 
772 
773 

88649 
88705 
88762 
88818 

88655 
88711 
88767 
88824 

88660 
88717 
88773 
88829 

88666 
88722 
88779 
88835 

88672 
88728 
88784 
88840 

88677 
88734 

88790 
88846 

88683 
88739 
88795 

88852 

88689 
88745 
88801 
88857 

88694 
88750 
88807 
88863 

88700  6 
88756 
88812 
88868 

774 
775 
776 

88874 
88930 
88986 

88880 
88936 
88992 

88885 
88941 
88997 

88891 
88947 
89003 

88897 
88953 
89009 

88902 
88958 
89014 

88908 
88964 
89020 

88913 
88969 
89025 

88919 
88975 
89031 

88925 
88981 
89037 

777 
778 
779 

89042 
89098 
89154 

89048 
89104 
89159 

89053 
89109 
89165 

89059 
89115 
89170 

89064 
89120 
89176 

89070 
89126 
89182 

89076 
89131 
89187 

89081 
89137 
89193 

89087 
89143 
89198 

89092 
89148 
89204 

780 

781 
782 
783 

89209 
89265 
89321 
89376 

89215 
89271 
89326 
89382 

89221 
89276 
89332 
8938J 

89226 
89282 
89337 
89393 

89232 
89287 
89343 
89398 

89237 
89293 
89348 
89404 

89243 
89298 
89354 
89409 

89248 
89304 
89360 
89415 

89254 
89310 
89365 
89421 

89260 
89315 
89371 
89426 

784 
785 

786 

89432 
.89487 
89542 

89437 
89492 
89548 

89443 
89498 
89553 

89448 
89504 
89559 

89454 
89509 
89564 

89459 
89515 
89570 

89465 
89520 
89575 

89470 
89526 
89581 

89476 
89531 
89586 

89481 
89537 
89592 

787 
788 
789 

89597 
89653 

89708 

89603 
89658 
89713 

89609 
89664 
89719 

89614 
89669 
89724 

89620 
89675 
89730 

89625 
89680 
89735 

89631 
89686 
89741 

89636 
89691 
89746 

89642 
89697 
89752 

89647 
89702 
89757 

790 

791 
792 
793 

89763 
89818 
89873 
89927 

89768 
89823 
89878 
89933 

89774 
89829 
89883 
89938 

89779 
89834 
89889 
89944 

89785 
89840 
89894 
89949 

89790 
89845 
89900 
89955 

89796 
89851 
89905 
89960 

89801 
89856 
89911 
89966 

89807 
89862 
89916 
89971 

89812 
89867 
89922 
89977 

794 
795 
796 

£9982 
90037 
90091 

89988 
90042 
90097 

89993 
90048 
90102 

89998 
90053 
90108 

90004 
90059 
90113 

90009 
90064 
90119 

90015 
90069 
•90124 

90020 
90075 
90129 

90026 
90080 
90135 

90031 
90086 
90140  5 

797 
798 
799 

90146 
90200 
90255 

90151 
90206 
90260 

90157 
90211 
90266 

90162 
90217 
90271 

90168 
90222 
90276 

90173 
90227 
90282 

90179 
90233 

90287 

90184 
90238 
90293 

90189 
90244 
90298 

90195 
90249 
90304 

COMMON    LOGARITHMS  27 

N0123456789D 

800  90309  90314  90320  90325  90331  90336  90342  90347  90352  90358 

801  90363  90369  90374  90380  90385  90390  90396  90401  90407  90412 

802  90417  90423  90428  90434  90439  90445  90450  90455  90461  90466 

803  90472  90477  90482  90488  90493  90499  90504  90509  90515  90520 

804  90526  90531  90536  90542  90547  90553  90558  90563  90569  90574 

805  90580  90585  90590  90596  90601  90607  90612  90617  90623  90628 

806  90634  90639  90644  90650  90655  90660  90666  90671  90677  90682 

807  90687  90693  90698  90703  90709  90714  90720  90725  90730  90736 

808  90741  90747  90752  90757  90763  90768  90773  90779  90784  90789 

809  90795  90800  90806  90811  90816  90822  90827  90832  90838  90843 

810  90849  90854  90859  90865  90870  90875  90881  90886  90891  90897 

811  90902  90907  90913  90918  90924  90929  90934  90940  90945  90950 

812  90956  90961  90966  90972  90977  90982  90988  90993  90998  91004 

813  91009  91014  91020  91025  91030  91036  91041  91046  91052  91057 

814  91062  91068  91073  9107S  91084  91089  91094  91100  91105  91110 

815  91116  91121  91126  91132  91137  91142  91148  91153  91158  91164 

816  91169  91174  91180  91185  91190  91196  91201  91206  91212  91217 

817  91222  91228  91233  91238  91243  91249  91254  91259  91265  91270 

818  91275  91281  91286  91291  91297  91302  91307  91312  91318  91323 

819  91328  91334  91339  91344  91350  91355  91360  91365  91371  91376 

820  91381  91387  91392  91397  91403  91408  91413  91418  91424  91429 

821  91434  91440  91445  91450  91455  91461  91466  91471  91477  91482 

822  91487  91492  91498  91503  91508  91514  91519  91524  91529  91535 

823  91540  91545  91551  91556  91561  91566  91572  91577  91582  91587 

824  91593  91598  91603  91609  91614  91619  91624  91630  91635  91640 

825  91645  91651  91656  91661  91666  91672  91677  91682  91687  91693  5 

826  91698  91703  91709  91714  91719  91724  91730  91735  91740  91745 

827  91751  91756  91761  91766  91772  91777  91782  91787  91793  91798 

828  91803  91808  91814  91819  91824  91829  91834  91840  91845  91850 

829  91855  91861  91866  91871  91876  91882  91887  91892  91897  91903 

830  91908  91913  91918  91924  91929  91934  91939  91944  91950  91955 

831  91960  91965  91971  91976  91981  91986  91991  91997  92002  92007 

832  92012  92018  92023  92028  92033  92038  92044  92049  92054  92059 

833  92065  92070  92075  92080  92085  92091  92096  92101  92106  92111 

834  92117  92122  92127  92132  92137  92143  92148  92153  92158  92163 

835  92169  92174  92179  92184  92189  92195  92200  92205  92210  92215 

836  92221  92226  92231  92236  92241  92247  92252  92257  92262  92267 

837  92273  92278  92283  92288  92293  92298  92304  92309  92314  92319 

838  92324  92330  92335  92340  92345  92350  92355  92361  92366  92371 

839  92376  92381  92387  92392  92397  92402  92407  92412  92418  92423 

840  92428  92433  92438  92443  92449  92454  92459  92464  92469  92474 

841  92480  92485  92490  92495  92500  92505  92511  92516  92521  92526 

842  92531  92536  92542  92547  92552  92557  92562  92567  92572  92578 

843  92583  92588  92593  92598  92603  92609  92614  92619  92624  92629 

844  92634  92639  92645  92650  92655  92660  92665  92670  92675  92681 

845  92686  92691  92696  92701  92706  92711  92716  92722  92727  92732 

846  92737  92742  92747  92752  92758  92763  92768  92773  92778  92783 

847  92788  92793  92799  92804  92809  92814  92819  92824  92829  92834 

848  92840  92845  92850  92855  92860  92865  92870  92875  92881  92886 

849  92891  92896  92901  92906  92911  92916  92921  92927  92932  92937 

N012345678  9    D 


28 

COMMON  LOGARITHMS 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9    D 

850 

851 
852 
853 

92942 
92993 
93044 
93095 

92947 
92998 
93049 
93100 

92952 
93003 
93054 
93105 

92957 
93008 
93059 
93110 

92962 
93013 
93064 
93115 

92967 
93018 
93069 
93120 

92973 
93024 
93075 
93125 

92978 
93029 
93080 
93131 

92983 
93034 
93085 
9313G 

92988 
93039 
930SO 
93141 

854 
855 

856 

93146 
93197 
93247 

93151 
93202 
93252 

93156 
93207 
93258 

93161 
93212 
93263 

93166 
93217 
93268 

93171 
93222 
93273 

93176 
93227 
93278 

93181 
93232 
93283 

93186 
93237 
93288 

93192 
93242 
93293 

857 
858 
859 

93298 
93349 
93399 

93303 
93354 
93404 

93308 
93359 
93409 

93313 
93364 
93414 

93318 
93369 
93420 

93323 
93374 
93425 

93328 
93379 
93430 

93334 
93384 
93435 

93339 
93389 
93440 

93344 
93394 
93445 

860 
861 
862 
863 

93450 
93500 
93551 
93601 

93455 
93505 
93556 
93606 

93460 
93510 
93561 
93611 

93465 
93515 
93566 
93616 

93470 
93520 
93571 
93621 

93475 
93526 
93576 
93626 

93480 
93531 
93581 
93631 

93485 
93536 
93586 
93636 

93490 
93541 
93591 
93641 

93495 
93546 
93596 
93646 

864 
865 
866 

93651 
93702 
93752 

93656 
93707 
93757 

93661 
93712 
93762 

93666 
93717 
93767 

93671 
93722 
93772 

93676 
93727 
93777 

93682 
93732 
93782 

93687 
93737 
93787 

93692 
93742 
93792 

93697 
93747 
93797 

867 
868 
869 

93802 
93852 
93902 

93807 
93857 
93907 

93812 
93862 
93912 

93817 
93867 
93917 

93822 
93872 
93922 

93827 
93877 
93927 

93832 
93882 
93932 

93837 
93887 
93937 

93842 
93892 
93942 

93847 
93897 
93947 

870 
871 
872 
873 

93952 
94002 
94052 
94101 

93957 
94007 
94057 
94106 

93962 
94012 
94062 
94111 

93967 
94017 
94067 
94116 

93972 
94022 
94072 
94121 

93977 
94027 
94077 
94126 

93982 
94032 
94082 
94131 

93987 
94037 
94086 
94136 

93992 
94042 
94091 
94141 

93997 
94047 
94096 
94146 

874 
875 
876 

94151 
94201 
94250 

94156 
94206 
94255 

94161 
94211 
94260 

94166 
94216 
94265 

94171 
94221 
94270 

94176 
94226 
94275 

94181 
94231 
94280 

96186 
94236 
94285 

94191 
94240 
94290 

94196 
94245  5 
94295 

877 
878 
879 

94300 
94349 
94399 

94305 
94354 
94404 

94310 
94359 
94409 

94315 
94364 
94414 

94320 
94369 
94419 

94325 
94374 
94424 

94330 
94379 
94429 

94335 
94384 
94433 

94340 
94389 
94438 

943*45 
94394 
94443 

880 

881 
882 
883 

94448 
94498 
94547 
94596 

94453 
94503 
94552 
94601 

94458 
94507 
94557 
94606 

94463 
94512 
94562 
94611 

94468 
94517 
94567 
94616 

94473 
94522 
94571 
94621 

94478 
94527 
94576 
94626 

94483 
94532 
94581 
94630 

94488 
94537 
94586 
94635 

94493 
94542 
94591 
94640 

884 
885 

886 

94645 
94694 
94743 

94650 
94699 
94748 

94655 
94704 
94753 

94660 
94709 
94758 

94665 
94714 
94763 

94670 
94719 
94768 

94675 
94724 
94773 

94680 
94729 
94778 

94685 
94734 
94783 

94689 
94738 
94787 

887 
888 
889 

94792 
94841 
94890 

94797 
94846 
94895 

94802 
94851 
94900 

94807 
94856 
94905 

94812 
94861 
94910 

94817 
94866 
94915 

94822 
94871 
94919 

94827 
94876 
94924 

94822 
94880 
94929 

94836 
94885 
94934 

890 

891 
892 
893 

94939 
94988 
95036 
95085 

94944 
94993 
95041 
95090 

94949 
94998 
95046 
95095 

94954 
95002 
95051 
95100 

94959 
95007 
95056 
95105 

94963 
95012 
95061 
95109 

94968 
95017 
95066 
95114 

94973 
95022 
95071 
95119 

94978 
95027 
95075 
95124 

94983 
95032 
95080 
95129 

894 
895 

896 

95134 
95182 
95231 

95139 
95187 
95236 

95143 
95192 
95240 

95148 
95197 
95245 

95153 
95202 
95250 

95158 
95207 
95255 

95163 
95211 
95260 

95168 
95216 
95265 

95173 
95221 
95270 

95177 
95226 
95274 

897 
898 
899 

95279 
95328 
95376 

95284 
95332 
95381 

95289 
95337 
95386 

95294 
95342 
95390 

95299 
95347 
95395 

95303 
95352 
95400 

95308 
95357 
95405 

95313 
95361 
95410 

95318 
95366 
95415 

95323 
95371 
95419 

COMMON  LOGARITHMS 

29 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9    D 

900 

901 
902 
903 

95424 
95472 
95521 
95569 

95429 
95477 
95525 
95574 

95434 
95482 
95530 
95578 

95439 
95487 
95535 
95583 

95444 
95492 
95540 
95588 

95448 
95497 
95545 
95593 

95453 
95501 
95550 
95598 

95458 
95506 
95554 
95602 

95463 
95511 
95559 

95607 

95468 
95516 
95564 
95612 

904 
905 
906 

95617 
95665 
95713 

95622 
95670 
95718 

95626 
95674 
95722 

95631 
95679 
95727 

95636 
95684 
95732 

95641 
95689 
95737 

95646 
95694 
95742 

95650 
95698 
95746 

95655 
95703 
95751 

95660 
95708 
95756 

907 
908 
909 

95761 
95809 
95856 

95766 
95813 
95861 

95770 
95818 
95866 

95775 
95823 
95871 

95780 
95828 
95875 

95785 
95832 
95880 

95789 
95837 
95885 

95794 
95842 
95890 

95799 
95847 
95895 

95804 
95852 
95899 

910 
911 
912 
913 

95904 
95952 
95999 
96047 

95909 
95957 
96004 
96052 

95914 
95961 
96009 
96057 

95918 
95966 
96014 
96061 

95923 
95971 
96019 
96066 

95928 
95976 
96023 
96071 

95933 
95980 
96028 
96076 

95938 
95985 
96033 
96080 

95942 
95990 
96038 
96085 

95947 
95995 
96042 
96090 

914 
915 
916 

96095 
96142 
96190 

96099 
96147 
96194 

96104 
96152 
96199 

96109 
96156 
96204 

96114 
96161 
96209 

96118 
96166 
96213 

96123 
96171 
96218 

96128 
96175 
96223 

96133 
96180 
96227 

96137 
96185 
96232 

917 
918 
919 

96237 
96284 
96332 

96242 
96289 
96336 

96246 
96294 
96341 

96251 
96298 
96346 

96256 
96303 
96350 

96261 
96308 
96355 

96265 
96313 
96360 

96270 
96317 
96365 

96275 
96322 
96359 

96280 
96327 
96374 

920 

921 
922 
923 

96379 
96426 
96473 
96520 

96384 
96431 
96478 
96525 

96388 
96435 
96483 
96530 

96393 
96440 
96487 
96534 

96398 
96445 
96492 
96539 

96402 
96450 
96497 
96544 

96407 
96454 
96501 
96548 

96412 
96459 
96506 
96553 

96417 
96464 
96511 
96558 

96421 
96468 
96515 
96562 

924 
925 

926 

96567 
96614 
96661 

96572 
96619 
96666 

96577 
96624 
96670 

96581 
96628 
96675 

96586 
96633 
96680 

96591 
96638 
96685 

96595 
96642 
96689 

96600 
96647 
96694 

96605 
96652 
96699 

96609   5 
96656 
96703 

927 
928 
929 

96708 
96755 
96802 

96713 
96759 
96806 

96717 
96764 
96811 

96722 
96769 
96816 

96727 
96774 
96820 

96731 
96778 
96825 

96736 
96783 
96830 

96741 
96788 
96834 

96745 
96792 
96839 

96750 
96797 
96844 

930 
931 
932 
933 

96848 
96895 
96942 
96988 

96853 
96900 
96946 
96993 

96858 
96904 
96951 
96997 

96862 
96909 
96956 
97002 

96867 
96914 
96960 
97007 

96872 
96918 
96965 
97011 

96876 
96923 
96970 
97016 

96881 
96928 
96974 
97021 

96886 
96932 
96979 
97025 

96890 
96937 
96984 
97030 

934 
935 

936 

97035 
97081 
97128 

97039 
97086 
97132 

97044 
97090 
97137 

97049 
97095 
97142 

97053 
97100 
97146 

97058 
97104 
97151 

97063 
97109 
97155 

97067 
97114 
97160 

97072 
97118 
97165 

97077 
97123 
97169 

937 

938 
939 

97174 
97220 
97267 

9717'9 
97225 
97271 

97183 
97230 
97276 

97188 
97234 

972.80 

97192 
97239 
97285 

97197 
97243 
97290 

97202 
97248 
97294 

97206 
97253 
97299 

97211 
97257 
97304 

97216 
97262 
97308 

940 
941 
942 
943 

97313 
97359 
97405 
97451 

97317 
97364 
97410 
97456 

97322 
97368 
97414 
97460 

97327 
97373 
97419 
97465 

97331 

97377 
97424 
97470 

97336 
97382 
97428 
97474 

97340 
97387 
97433 
97479 

97345 
97391 
97437 
97483 

97350 
97396 
97442 

97488 

97354 
97400 
97447 
97493 

944 
945 
946 

97497 
97543 
97589 

97502 
97548 
97594 

97506 
97552 
97598 

97511 
97557 
97603 

97516 
97562 
97607 

97520 
97566 
97612 

97525 

97571 
97617 

97529 
97575 
97621 

97534 
97580 
97626 

97539 
97585 
97630 

947 

948 
949 

97635 
97681 

97727 

97640 
97685 
97731 

97644 
97690 
97736 

97649 
97695 
97740 

97653 
97699 
97745 

97658 
97704 
97749 

97663 
97708 
97754 

97667 
97713 
97759 

97672 
97717 
97763 

97676 
97722 
97768 

30 


COMMON    LOGARITHMS 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 

950 
951 
952 
953 

97772 
97818 
97864 
97909 

97777 
97823 
97868 
97914 

97782 
97827 
97873 
97918 

97786 
97832 
97877 
97923 

97791 
97836 
97882 
97928 

97795 
97841 
97886 
97932 

.97800 
97845 
97891 
97937 

97804 
97850 
97896 
97941 

97809 
97855 
97900 
97946 

97813 
97859 
97905 
97950 

954 
955. 

956 

97955 
98000 
98046 

97959 
98005 
98050 

97964 
98009 
98055 

97968 
98014 
98059 

97973 
98019 
98064 

97978 
98023 
98068 

97982 
98028 
98073 

97987 
98032 
98078 

97991 
98037 
98082 

97996 
98041 
98087 

957 
958 
959 

98091 
98137 
98182 

98096 
98141 

98186 

98100 
98146 
98191 

98105 
98150 
98195 

98109 
98155 
98200 

98114 
98159 

98204 

98118 
98164 
98209 

98123 
98168 
98214 

98127 
98173 
98218 

98132 
98177 
98223 

5 

960 
961 
962 
963 

98227 
98272 
98318 
98363 

98232 
98277 
98322 
98367 

98236 
98281 
98327 
98372 

98241 
98286 
98331 
98376 

98245 
98290 
98336 
98381 

98250 
98295 
98340 
98385 

98254 
98299 
98345 
98390 

98259 
98304 
98349 
98394 

98263 
98308 
98354 
98399 

98268 
98313 
98358 
98403 

964 

98408 

98412 

98417 

98421 

98426 

98430 

98435 

98439 

98444 

98448 

965 

966 

98453 

98498 

98457 
98502 

98462 

98507 

98466 
98511 

98471 
98516 

98475 
98520 

98480 
98525 

98484 
98529 

98489 
98534 

98493 
98538 

967 

968 
969 

98543 
98588 
98632 

98547 
98592 
98637 

98552 
98597 
98641 

98556 
98601 
98646 

98561 
98605 
98650 

98565 
98610 
98655 

98570 
98614 
98659 

98574 
98619 
98664 

98579 
98623 

98668 

98583 
98628 
98673 

970 

971 
972 
973 

98677 
98722 
98767 
98811 

98682 
98726 
98771 
98816 

98686 
98731 
98776 
98820 

98691 
98735 
98780 
98825 

98695 
98740 
98784 
98829 

98700 
98744 
98789 
98834 

98704 
98749 
98793 
98838 

98709 
98753 
98798 
98843 

98713 

98758 
98802 
98847 

98717 
98762 
98807 
98851 

974 
975 
976 

98856 
98900 
98945 

98860 
98905 
98949 

98865 
98909 
98954 

98869 
98914 
98958 

98874 
98918 
98963 

98878 
98923 
98967 

98883 
98927 
98972 

98887 
98932 
98976 

98892 
98936 
98981 

98896 
98941 

98985 

977 

978 
979 

98989 
99034 
99078 

98994 
99038 
99083 

98998 
99043 

99087 

99003 
99047 
99092 

99007 
99052 
99096 

99012 
99056 
99100 

99016 
99061 
99105 

99021 
99065 
99109 

99025 
99069 
99114 

99029 
99074 
99118 

980 

981 
982 
983 

99123 
99167 
99211 
99255 

99127 
99171 
99216 
99260 

99131 
99176 
99220 
99264 

99136 
99180 
99224 
99269 

99140 
99185 
99229 
99273 

99145 
99189 
99233 
99277 

99149 
99193 
99238 
99282 

99154 
99198 
99242 
99286 

99158 
99202 
99247 
99291 

99162 
99207 
99251 
99295 

4 

984 
985 
986 

99300 
99344 
99388 

99304 
99348 
99392 

99308 
99352 
99396 

99313 
99357 
99401 

99317 
99361 
99405 

99322 
99366 
99410 

99326 
99370 
99414 

99330 
99374 
99419 

99335 
99379 
99423 

99339 
99383 
99427 

987 
988 
989 

99432 
99476 
99520 

99436 
99480 
99524 

99441 
99484 
99528 

99445 
99489 
99533 

99449 
99493 
99537 

99454 
99498 
99542 

99458 
99502 
99546 

99463 
99506 
99550 

99467 
99511 
99555 

99471 
99515 
99559 

990 

991 
992 
993 

99564 
99607 
99651 
99695 

99568 
99612 
99656 
99699 

99572 
99616 
99660 
99704 

99577 
99621 
99664 

99708 

99581 
99625 
99669 
99712 

99585 
99629 
99673 
99717 

99590 
99634 
99677 
99721 

99594 
99638 

99682 
99726 

99599 
99642 
99686 
99730 

99603 
99647 
99691 
99734 

994 
995 

996 

99739 
99782 
99826 

99743 
99787 
99830 

99747 
99791 
99835 

99752 
99795 
9983D 

99756 
99800 
99843 

99760 
99804 
99848 

99765 
99808 
99852 

99769 
99813 
99856 

99774 
99817 
99861 

99778 
99822 
99865 

997 
998 
999 

99870 
99913 
99957 

99874 
99917 
99961 

99878 
99922 
99965 

99883 
99926 
99970 

99887 
99930 
99974 

99891 
99935 
99978 

99896 
99939 
99983 

99900 
99944 
99987 

99904 
99948 
99991 

99909 
99952 
99996 

TRIGONOMETRIC    FUNCTIONS 


Sine 

Tangent 

Cotangent 

Cosine 

Ang, 

.  Nat. 

Log. 

Nat. 

Log. 

Nat. 

Log. 

Nat. 

Log. 

7 

7 

2 

9 

0 

0 

0 

1.00000 

0 

90 

.1 

.00175 

.24188 

.00175 

.24188 

572.96 

.75812 

1.00000 

.99999 

.9 

.2 

.00349 

.54291 

.00349 

.54291 

286.48 

.45709 

.99999 

.99999 

.8 

.3 

.00524 

.71900 

.00524 

.71900 

190.98 

.28100 

.99999 

.99999 

.7 

.4 

.00698 

.84393 

.00698 

.84394 

143.24 

.15606 

.99998 

.99999 

.6 

.5 

.00873 

.94084 

.00873 

.94086 

114.59 

.05914 

.99996 

.99998 

.5 

8 

8 

1 

9 

.6 

.01047 

.02002 

.01047 

.02004 

95.489 

.97996 

.99995 

.99998 

.4 

.7 

.01222 

.08697 

.01222 

.08700 

81.847 

.91300 

.99993 

.99997 

.3 

.8 

.01396 

-.14495 

.01396 

.14500 

71.615 

.85500 

.99990 

.99996 

.2 

.9 

.01571 

.19610 

.01571 

.19616 

63.657 

.80384 

.99988 

.99995 

.1 

1.0 

.01745 

.24186 

.01746 

.  24192 

57.290 

.75808 

.99985 

.99993 

89 

.1 

.01920 

.28324 

.01920 

.28332 

52.081 

.71668 

.99982 

.99992 

9 

.2 

.02094 

.32103 

.02095 

.32112 

47.740 

.67888 

.99978 

.99990 

.8 

.3 

.02269 

.35578 

.02269 

.35590 

44.066 

.64410 

.99974 

.99989 

.7 

.4 

.02443 

.38796 

.02444 

.38809 

40.917 

.61191 

.99970 

.99987 

.6 

.5 

.02618 

.41792 

.02619 

.41807 

38.188 

.58193 

.99966 

.99985 

5 

.6 

.02792 

.44594 

.02793 

.44611 

35.801 

.55389 

.99961 

.99983 

.4 

.7 

.02967 

.47226 

.02968 

.47245 

33.694 

.52755 

.99956 

.99981 

.3 

.8 

.03141 

.49708 

.03143 

.49729 

31.821 

.50271 

.99951 

.99979 

2 

.9 

.03316 

.52055 

.03317 

.52079 

30.145 

.47921 

.99945 

.99976 

!i 

2.0 

.03490 

.54282 

.03492 

.54308 

28.636 

.45692 

.99939 

.99974 

88 

.1 

.03664 

.56400 

.03667 

.56429 

27.271 

.43571 

.99933 

.99971 

.9 

2 

.03839 

.58419 

.03842 

.58451 

26.031 

.41549 

.99926 

.99968 

.8 

!3 

.04013 

.60349 

.01016 

.60384 

24.898 

.39616 

.99919 

.99965 

.7 

.4 

.04188 

.62196 

.04191 

.62234 

23.859 

.37766 

.99912 

.99962 

.6 

.5 

.04362 

.63968 

.04366 

.64009 

22.904 

.35991 

.99905 

.99959 

.5 

.6 

.04536 

.65670 

.04541 

.65715 

22.022 

.34285 

.99897 

.99955 

.4 

.7 

.04711 

.67308 

.04716 

.67356 

21.205 

.32644 

.99889 

.99952 

.3 

.8 

.04885 

.68886 

.04891 

.68938 

20.446 

.31062 

.99881 

.99948 

.2 

.9 

.05059 

.70409 

.05066 

.70465 

19.740 

.29535 

.99872 

.99944 

.1 

3.0 

.05234 

.71880 

.05241 

.71940 

19.081 

.28060 

.99863 

.99940 

87 

.1 

.05408 

.73303 

.05416 

.73366 

18.464 

.26634 

.99854 

.99936 

.9 

2 

.05582 

.74680 

.05591 

.74748 

17.886 

.25252 

.99844 

.99932 

.8 

!s 

.05756 

.76015 

.05766 

.76087 

17.343 

.23913 

.98334 

.99928 

.7 

.4 

.05931 

.77310 

.05941 

.77387 

16.832 

.22613 

.99824 

.99923 

.6 

.5 

.06105 

.78568 

.06116 

.78649 

16.350 

.21351 

.99813 

.99919 

.5 

.6 

.06279 

.79789 

.06291 

.79875 

15.895 

.20125 

.99803 

.99914 

.4 

.7 

.06453 

.80978 

.06467 

.81068 

15.464 

.18932 

.99792 

.99909 

.3 

.8 

.06627 

.82134 

.06642 

.82230 

15.056 

.17770 

.99780 

.99904 

.2 

.9 

.06802 

.83261 

.06817 

.83361 

14.669 

.16639 

.99768 

.99899 

.1 

4.0 

.06976 

.84358 

.06993 

.84464 

14.301 

.15536 

.99756 

.99894 

86 

.1 

.07150 

.85429 

.07168 

.85540 

13.951 

.14460 

.99744 

.99889 

.9 

2 

.07324 

.86474 

.07344 

.86591 

13.617 

.13409 

.99731 

.99883 

.8 

!s 

.07498 

.87494 

.07519 

.87616 

13.300 

.12384 

.99719 

.99878 

.7 

.4 

.07672 

.88490 

.07695 

.88619 

12.996 

.11382 

.99705 

.99872 

.6 

.5 

.07846 

.89464 

.07870 

.89598 

12.706 

.10402 

.99692 

.99866 

.5 

.6 

.08020 

.90417 

.08046 

.90557 

12.429 

.09443 

.99678 

.99860 

.4 

.7 

.08194 

.91349 

.08221 

.91495 

12.163 

.08505 

.99664 

.99854 

.3 

.8 

.08368 

.92261 

.08397 

.92414 

11.909 

.07586 

.99649 

.99847 

.2 

.9 

.08542 

.93154 

.08573 

.93313 

11.664 

.06687 

.99635 

.99841 

.1 

5.0 

.08716 

.94030 

.08749 

.94195 

11.430 

.05805 

.99619 

.99834 

85 

Nat. 

Log. 

Nat. 

Log. 

Nat. 

Log. 

Nat. 

Log.Ang. 

Cosine 

Cotangent 

Tangent 

Sine 

TRIGONOMETRIC    FUNCTIONS 


Sine 

Tangent 

Cotangent 

Cosine 

Ang 

.  Nat. 

Log. 

Nat. 

Log. 

Nat. 

Log. 

Nat. 

Log. 

8 

8 

1 

9 

5.0 

.08716 

.94030 

.08749 

.04195 

11.430 

.05805 

.99619 

.99834 

85 

.1 

.08889 

.94887 

.08925 

.95060 

11.205 

.04940 

.99604 

.99828 

.9 

.2 

.09063 

.95728 

.09101 

.95908 

10.988 

.04092 

.99588 

.99821 

.8 

.3 

.09237 

.96553 

.09277 

.96739 

10.780 

.03261 

.99572 

.99814 

.7 

.4 

.09411 

.97363 

.09453 

.97556 

10.579 

.02444 

.99556 

.99807 

.6 

.5 

.09585 

.98157 

.09629 

.98358 

10.385 

.01642 

.99540 

.99800 

.5 

.6 

.09758 

.98937 

.09805 

.99145 

10.199 

.00855 

.99523 

.99792 

.4 

.7 

.09932 

.99704 

.09981 

.99919 

10.019 

.00081 

.99506 

.99785 

.3 

9 

9 

10 

9 

.8 

.10106 

.00456 

.10158 

.00679 

9.4480 

.99321 

.99488 

.99777 

.2 

.9 

.10279 

.01196 

.10334 

.01427 

9.6768 

.98573 

.99470 

.99769 

.1 

6.0 

.10453 

.01923 

.10510 

.02162 

9.5144 

.97838 

.99452 

.99761 

84 

.1 

.10626 

.02639 

.10687 

.02885 

9.3572 

.97115 

.99434 

.99753 

.9 

.2 

.10800 

.03342 

.10863 

.03597 

9.2052 

.96403 

.99415 

.99745 

.8 

.3 

.10973 

.04034 

.11040 

.04297 

9.0579 

.95703 

.99396 

.99737 

.7 

.4 

.11147 

.04715 

.11217 

.04987 

8.9152 

.95013 

.99377 

.99729 

.6 

.5 

.11320 

.05386 

.11394 

.05666 

8.7769 

.94334 

.99357 

.99720 

.5 

.6 

.11494 

.06046 

.11570 

.06335 

8.6427 

.93665 

.99337 

.99711 

.4 

.7 

.11667 

.06696 

.11747 

.06994 

8.5126 

.93006 

.99317 

.99702 

.3 

.8 

.11840 

.07337 

.11924 

.07643 

8.3863 

.92357 

.99297 

.99693 

.2 

.9 

.12014 

.07968 

.12101 

.08283 

8.2636 

.91717 

.99276 

.99684 

.1 

7.0 

.12187 

.08589 

.12278 

.08914 

8.1443 

.91086 

.99255 

.99675 

83 

.1 

.12360 

.09202 

.12456 

.09537 

8.0285 

.90463 

.99233 

.99666 

.3 

.2 

.12533 

.09807 

.12633 

.10150 

7.9158 

.89850 

.99211 

.99656 

.8 

.3 

.12706 

.10403 

.12810 

.10756 

7.8062 

.89244 

.99189 

.99647 

.7 

.4 

.12880 

.10990 

.12988 

.11353 

7.6996 

.88647 

.99167 

.99637 

.6 

.5 

.13053 

.11570 

.13165 

.11943 

7.5958 

.88057 

.99144 

.99627 

.5 

.6 

.13226 

.12142 

.13343 

.12525 

7.4947 

.87475 

.99122 

.99617 

.4 

.7 

.13399 

.12706 

.13521 

.13099 

7.3962 

.86901 

.99098 

.99607 

.3 

.8 

.13572 

.13263 

.13698 

.13667 

7.3002 

.86333 

.99075 

.99596 

.2 

.9 

.13744 

.13813 

.13876 

.14227 

7.2066 

.85773 

.99051 

.99586 

.1 

8.0 

.13917 

.14356 

.14054 

.14780 

7.1154 

.85220 

.99027 

.99575 

82 

.1 

.14090 

.14891 

.14232 

.15327 

7.0264 

.84673 

.99002 

.99565 

.9 

.2 

.14263 

.15421 

.14410 

.15867 

6.9395 

.84133 

.98978 

.99554 

.8 

.3 

.14436 

.15944 

.14588 

.16401 

6.8548 

.83599 

.98953 

.99543 

.7 

.4 

.14608 

.16460 

.14767 

.16928 

6.7720 

.83072 

.98927 

.99532 

.6 

.5 

.14781 

.16970 

.14945 

.17450 

6.6912 

.82550 

.98902 

.99520 

.5 

.6 

.14954 

.17474 

.15124 

.17965 

6.6122 

.82035 

.98876 

.99509 

.4 

.7 

.15126 

.17973 

.15302 

.18475 

6.5350 

.81525 

.98849 

.99497 

.3 

.8 

.15299 

.18465 

.15481 

.18979 

6.4596 

.81021 

.98823 

.99486 

.2 

.9 

.15471 

.18952 

.15660 

.19478 

6.3859 

.80522 

.98796 

.99474. 

.1 

9.0 

.15643 

.19433 

.15838 

.19971 

6.3138 

.80029 

.98769 

.99462 

81 

.1 

.15816 

.19909 

.16017 

.20459 

6.2432 

.79541 

.98741 

.99450 

.9 

.2 

.15988 

.20380 

.16196 

.20942 

6.1742 

.79058 

.98714 

.99438 

.8 

.3 

.16160 

.20845 

.16376 

.21420 

6.1066 

.78580 

.98686 

.99425 

.7 

.4 

.16333 

.21306 

.16555 

.21893 

6.0405 

.78107 

.98657 

.99413 

.6 

.5 

.16505 

.21761 

.16734 

.22361 

5.9758 

.77639 

.98629 

.99400 

.5 

.6 

.16677 

.22211 

.16914 

.  22824 

5.9124 

.77176 

.98600 

.99388 

.4 

.7 

.16849 

.22657 

.17093 

.  23283 

5.8502 

.76717 

.98570 

.99375 

.3 

.8 

.17021 

.23098 

.17273 

.23737 

5.7894 

.76263 

.98541 

.99362 

2 

.9 

.17193 

.23535 

.17453 

.24187 

5.7297 

.75814 

.98511 

.99348 

!i 

10 

.17365 

.23967 

.17633 

.24632 

5.6713 

.75368 

.98481 

.99335 

80 

Nat. 

Log. 

Nat. 

Log. 

Nat. 

Log. 

Nat. 

Log.Ang. 

Cosine 

Cotangent 

Tangent 

Sine 

TRIGONOMETRIC    FUNCTIONS 


33 


Sine 

Tangent 

Cotangent 

Cosine 

Ang 

Nat. 

Log. 

Nat. 

Log. 

Nat. 

Log. 

Nat. 

Log. 

9 

9 

10 

9 

10 

.17365 

.23967 

.17633 

.24632 

5.6713 

.75368 

.98481 

.99335 

80 

.1 

.17537 

.24395 

.17813 

.25073 

5.6140 

.74927 

.98450 

.99322 

.9 

2 

.17708 

.24818 

.17993 

.25510 

5.5578 

.74490 

.98420 

.99308 

.8 

.3 

.17880 

.25237 

.18173 

.25943 

5.5026 

.74057 

.98389 

.99294 

.7 

.4 

.18052 

.25652 

.18353 

.26372 

5.4486 

.73628 

.98357 

.99281 

.6 

.5 

.18224 

.26063 

.18534 

.26797 

5.3955 

.73203 

.98325 

.99267 

.5 

.6 

.18395 

.26470 

.18714 

.27218 

5.3435 

.72782 

.98294 

.99253 

.4 

.7 

.18567 

.26873 

.18895 

.27635 

5.2924 

.72365 

.98261 

.99238 

.3 

.8 

.18738 

.27273 

.19076 

.28049 

5.2422 

.71951 

.98229 

.99224 

.2 

.9 

.18910 

.27668 

.19257 

.28459 

5.1929 

.71541 

.98196 

.99209 

.1 

11 

.19081 

.28060 

.19438 

.28865 

5.1446 

.71135 

.98163 

.99195 

79 

.1 

.19252 

.28448 

.19619 

.29268 

5.0970 

.70732 

.98129 

.99180 

.9 

2 

.19423 

.28833 

.19801 

.29668 

5.0504 

.70332 

.98096 

.99165 

.8 

.'3 

.19595 

.29214 

.19982 

.30064 

5.0045 

.69936 

.98061 

.99150 

.7 

.4 

.19766 

.29591 

.20164 

.30457 

4.9594 

.69543 

.98027 

.99135 

.6 

.5 

.19937 

.29966 

.20345 

.30846 

4.9152 

.69154 

.97992 

.99119 

.5 

.6 

.20108 

.30336 

.20527 

.31233 

4.8716 

.68767 

.97958 

.99104 

.4 

.7 

.20279 

.30704 

.20709 

.31616 

4.8288 

.68384 

.97922 

.99088 

.3 

.8 

.20450 

.31068 

.20891 

.31996 

4.7867 

.68004 

.97887 

.99072 

.2 

.9 

.20620 

.31430 

.21073 

.32373 

4.7453 

.67627 

.97851 

.99056 

.1 

12 

.20791 

.31788 

.21256 

.32747 

.7046 

.67253 

.97815 

.99040 

.  78 

.1 

.20962 

.32143 

.21438 

.33119 

.6646 

.66881 

.97778 

.99024 

.9 

.2 

.21132 

.32495 

.21621 

.33487 

.6252 

.66513 

.97742 

.99008 

.8 

.3 

.21303 

.32844 

.21804 

.33853 

.5864 

.66147 

.97705 

.98991 

.7 

.4 

.21474 

.33190 

.21986 

.34215 

.5483 

.65785 

.97667 

.98975 

.6 

.5 

.21644 

.33534 

.21169 

.34576 

.5107 

.65424 

.97630 

.98958 

.5 

.6 

.21814 

.33874 

.22353 

.34933 

.4737 

.65067 

.97592 

.98941 

.4 

.7 

.21985 

.34212 

.22536 

.35288 

4.4373 

.64712 

.97553 

.98924 

.3 

.8 

.22155 

.34547 

.22719 

.35640 

4.4015 

.64360 

.97515 

.98907 

.2 

.9 

.22325 

.34879 

.22903 

.35989 

4.3662 

.64011 

.97476 

.98890 

.1 

13 

.22495 

.35209 

.23087 

.36336 

4.3315 

.63664 

.97437 

.98872 

77 

.1 

.22665 

.35536 

.23271 

.36681 

4.2972 

.63319 

.97398 

.98855 

.9 

.2 

.22835 

.35860 

.23455 

.37023 

4.2635 

.62977 

.97358 

.98837 

.8 

.3 

.23005 

.36182 

.23639 

.37363 

.2303 

.62637 

.97318 

.98819 

.7 

.4 

.23175 

.36502 

.23823 

.37700 

.1976 

.62300 

.97278 

.98801 

.6 

.5 

.23345 

.36819 

.24008 

.38035 

.1653 

.61965 

.97237 

.98783 

.5 

.6 

.23514 

.3/133 

.24193 

.38368 

.1335 

.61632 

.97196 

.98765 

.4 

.7 

.23684 

.37445 

.24377 

.38699 

.1022 

.61301 

.97155 

.98746 

.3 

.8 

.23853 

.37755 

.24562 

.39027 

.0713 

.60973 

.97113 

.98728 

.2 

.9 

.24023 

.38062 

.24747 

.39353 

.0408 

.60647 

.97072 

.98T09 

.1 

14 

.24192 

.38368 

.24933 

.39677 

4.0108 

.60323 

.97030 

.98690 

76 

.1 

.24362 

.38670 

.25118 

.39999 

3.9812 

.60001 

.96987 

.98671 

.9 

2 

.24531 

.38971 

.25304 

.40319 

3.9520 

.59681 

.96945 

.98652 

.8 

'.3 

.24700 

.39270 

.25490 

.40636 

3  9232 

.59364 

.96902 

.98633 

.7 

.4 

.24869 

.39566 

.25676 

.40952 

s!8947 

.59048 

.96858 

.98614 

-.6 

.5 

.25038 

.39860 

.25862 

.41266 

3.8667 

.58734 

.96815 

.98594 

.5 

.6 

.25207 

.40152 

.26048 

.41578 

3.8391 

.58422 

.96771 

.98574 

.4 

.7 

.25376 

.40442 

.26235 

.41887 

3.8118 

.58113 

.96727 

.98555 

.3 

.8 

.25545 

.40730 

.26421 

.42195 

3.7848 

.57805 

.96682 

.98535 

2 

.9 

.25713 

.41016 

.26608 

.42501 

3.7583 

.57499 

.96638 

.98515 

.'l 

15 

.25882 

.41300 

.26795 

.42805 

3.7321 

.57195 

.96593 

.98494 

75 

Nat. 

LOST. 

Nat. 

Log. 

Nat. 

Log. 

Nat. 

Log.Ang. 

Cosine 

Cotangent 

Tangent 

Sine 

34 


TRIGONOMETRIC    FUNCTIONS 


Sine 

Tangent 

Cotangent 

Cosine 

Ang. 

,  Nat. 

Log. 

Nat. 

Log. 

Nat. 

Log. 

Nat. 

Log. 

9 

9 

10 

9 

15 

.25882 

.41300 

.26795 

.42805 

3.7321 

.57195 

.96593 

.98494 

75 

.1 

.26050 

.41582 

.26982 

.43108 

3.7062 

.56892 

.96547 

.98474 

.9 

.2 

.26219 

.41861 

.27169 

.43408 

3.6806 

.56592 

.96502 

.98453 

.8 

.3 

.26387 

.42140 

.27357 

.43707 

3.6554 

.56293 

.96456 

.98433 

.7 

.4 

.26556 

.42416 

.27545 

.44004 

3.6305 

.55996 

.96410 

.98412 

.6 

.5 

.26724 

.42690 

.27732 

.44299 

3.6059 

.55701 

.96363 

.98391 

.5 

.6 

.26892 

.42962 

.27921 

.44592 

3.5816 

.55408 

.96316 

.98370 

.4 

.7 

.27060 

.43233 

.28109 

.44884 

3.5576 

.55116 

.96269 

.98349 

.3 

.8 

.27228 

.43502 

.28297 

.45174 

3.5339 

.54826 

.96222 

.98327 

.2 

.9 

.27396 

.43769 

.28486 

.45463 

3.5105 

.54537 

.96174 

.98306 

.1 

16 

.27564 

.44034 

.28675 

.45750 

3.4874 

.54250 

.96126 

.98284 

74 

.1 

.27731 

.44297 

.28864 

.46035 

3.4646 

.53965 

.96078 

.98262 

.9 

.2 

.27899 

.44559 

.29053 

.46319 

3.4420 

.53681 

.96029 

.98240 

.8 

.3 

.28067 

.44819 

.29242 

.46601 

3.4197 

.53399 

.95981 

.98218 

.7 

A 

.28234 

.45077 

.29432 

.46881 

3.3977 

.53119 

.95931 

.98196 

.6 

.5 

.28402 

.45334 

.29621 

.47161 

3.3759 

.52840 

.95882 

.98174 

.5 

.6 

.28569 

.45589 

.29811 

.47438 

3.3544 

.52562 

.95832 

.98151 

.4 

.7 

.28736 

.45843 

.30001 

.47714 

3.3332 

.52286 

.95782 

.98129 

.3 

.8 

.  2890? 

.46095 

.30192 

.47989 

3.3122 

.52011 

.95732 

.98106 

.2 

.9 

.29070 

.46345 

.30382 

.48262 

3.2914 

.51738 

.95681 

.98083 

.1 

17. 

.29237 

.46594 

.30573 

.48534 

3.2709 

.51466 

.95630 

.98060 

73 

.1 

.29404 

.46841 

.3076* 

.48804 

3.2506 

.51196 

.95579 

.98036 

.9 

.2 

.29571 

.47086 

.30955 

.49073 

3.2305 

.50927 

.95528 

.98013 

.8 

.3 

.29737 

.47330 

.31147 

.49341 

3.2106 

.50659 

.95476 

.97989 

.7 

.4 

.29904 

.47573 

.31338 

.49607 

3.1910 

.50393 

.95424 

.97966 

.6 

.5 

.30071 

.47814 

.31530 

.49872 

3.1716 

.50128 

.95372 

.97942 

.5 

.6 

.30237 

.48054 

.31722 

.50136 

3.1524 

.49864 

.95319 

.97918 

.4 

.7 

.30403 

.48292 

.31914 

.50398 

3.1334 

.49602 

.95266 

.97894 

.3 

.8 

.30570 

.48529 

.32106 

.50659 

3.1146 

.49341 

.95*213 

.97870 

.2 

.y 

.30736 

.48764 

.32299 

.50919 

3.0961 

.49081 

.95159 

.97845 

.1 

18 

.30902 

.48998 

.32492 

.51178 

3.0777 

.48822 

.95106 

.97821 

72 

.1 

.31068 

.49231 

.32685 

.51435 

3.0595 

.48565 

.95052 

.97796 

.9 

.2 

.31233 

.49462 

.32878 

.51691 

3.0415 

.48309 

.94997 

.97771 

.8 

.3 

.31399 

.49692 

.33072 

.51946 

3.0237 

.48054 

.94943 

.97746 

.7 

.4 

.31565 

.49920 

.33266 

.52200 

3.0061 

.47801 

.94888 

.97721 

.6 

.5 

.31730 

.50148 

.33460 

.52452 

2.9887 

.47548 

.94832 

.97696 

.5 

.6 

.31896 

.50374 

.33654 

.52703 

2.9714 

.47297 

.94777 

.97670 

.4 

.7 

.32061 

.50598 

.33348 

.52953 

2.9544 

.47047 

.94721 

.97645 

.3 

.8 

.32227 

.50821 

.34043 

.53203 

2.9375 

.46798 

.94665 

.97619 

.2 

.9 

.32392 

.51043 

.34238 

.53450 

2.9208 

.46550 

.94609 

.97593 

.1 

19 

.32557 

.51264 

.34433 

.53697 

2.9042 

.46303 

.94552 

.97567 

71 

.1 

.32722 

.51484 

.34628 

.53943 

2.8878 

.46057 

.94495 

.97541 

.9 

.2 

.32887 

.51702' 

.34824 

.54187 

2.8716 

.45813 

.94438 

.97515 

.8 

.3 

.33051 

.51919 

.35020 

.54431 

2.8556 

.45569 

.94380 

.97488 

.7 

.4^ 

.33216 

.52135 

.35216 

.54673 

2.8397 

.45327 

.94322 

.97461 

.6 

.5 

.33381 

.52350 

.35412 

.54915 

2.8239 

.45085 

.94264 

.97435 

.5 

.6 

.33545 

.52563 

.35608 

.55155 

2.8083 

.44845 

.94206 

.97408 

.4 

.7 

.33710 

.52775 

.35805 

.55395 

2.7929 

.44605 

.94147 

.97381 

.3 

.8 

.33874 

.52986 

.36002 

.55633 

2.7776 

.44367 

.94088 

.97353 

.2 

.9 

.34038 

.53196 

.36199 

.55870 

2.7625 

.44130 

.94029 

.97326 

.1 

20 

.34202 

.53405 

.36397 

.56107 

2.7475 

.43893 

.93969 

.97299 

70 

Nat. 

Log. 

Nat. 

Log. 

Nat. 

Log. 

Nat. 

Log.Ang. 

Cosine 

Cotangent 

Tangent 

Sine 

TRIGONOMETRIC    FUNCTIONS 


35 


Sine 

Tangent 

Cotangent 

Cosine 

Ang, 

.  Nat. 

Log. 

Nat. 

Log. 

Nat. 

Log. 

Nat. 

Log. 

9 

9 

10 

9 

20 

.34202 

.53405 

.36397 

.56107 

2.7475 

.43893 

.93969 

.97299 

70 

.1 

.34366 

.53613 

.36595 

.56342 

2.7326 

.43658 

.93909 

.97271 

.9 

.2 

.34530 

.53819 

.36793 

.56576 

2.7179 

.43424 

.93849 

.97243 

.8 

.3 

.34694 

.54025 

.36991 

.56810 

2.7034 

.43190 

.93789 

.97215 

.7 

.4 

.34857 

.54229 

.37190 

.57042 

2.6889 

.42958 

.93728 

.97187 

.6 

.5 

.35021 

.54433 

.37388 

.57274 

2.6746 

.42726 

.93667 

.97159 

.5 

.6 

.35184 

.54635 

.37588 

.57504 

2.6605 

.42496 

.93606 

.97130 

.4 

.7 

.35347 

.54836 

.37787 

.57734 

2.6464 

.42266 

.93544 

.97102 

.3 

.8 

.35511 

.55036 

.37986 

.57963 

2.6325 

.42037 

.93483 

.97073 

.2 

.9 

.35674 

.55235 

.38186 

.58191 

2.6187 

.41809 

.93420 

.97044 

.1 

21 

.35837 

.55433 

.38386 

.58418 

2.6051 

.41582 

.93358 

.97015 

69 

.1 

.36000 

.55630 

.38587 

.58644 

2.5916 

.41356 

.93295 

.96986 

.9 

.2 

.36162 

.55826 

.38787 

.58869 

2.5782 

.41131 

.93232 

.96957 

.8 

.3 

.36325 

.56021 

.38988 

.59094 

2.5649 

.40906 

.93169 

.96927 

.7 

.4 

.36488 

.56215 

.39190 

.59317 

2.5517 

.40683 

.93106 

.96898 

.6 

.5 

.36650 

.56408 

.39391 

.59540 

2.5386 

.40460 

.93042 

.96868 

.5 

.6 

.36812 

.56600 

.39593 

.59762 

2.5257 

.40238 

.92978 

.96838 

.4 

.7 

.36975 

.56790 

.39795 

.59983 

2.5129 

.40017 

.92913 

.96808 

.3 

8 

.37137 

.56980 

.39997 

.60203 

2.5002 

.39797 

.92849 

.96778 

.2 

.9 

.37299 

.57169 

.40200 

.60422 

2.4876 

.39578 

.92784 

.96747 

.1 

22 

.37461 

.57358 

.40403 

.60641 

2.4751 

.39359 

.92718 

.96717 

68 

.1 

.37622 

.57545 

.40606 

.60859 

2.4627 

.39141 

.92653 

.96686 

.9 

.2 

.37784 

.57731 

.40809 

.61076 

2.4504 

.38924 

.92587 

.96655 

.8 

.3 

.37946 

.57916 

.41013 

.61292 

2.4383 

.38708 

.92521 

.96624 

.7 

.4 

.38107 

.58101 

.41217 

.61508 

2.4262 

.38492 

.92455 

.96593 

.6 

.5 

.38268 

.58284 

.41421 

.61722 

2.4142 

.38278 

.92388 

.96562 

.5 

.6 

.38430 

.58467 

.41626 

.61936 

2.4023 

.38064 

.92321 

.96530 

.4 

.7 

.38591 

.58648 

.41831 

.62150 

2.3906 

.37850 

.92254 

.96498 

.$ 

.8 

.38752 

.58829 

.42036 

.62362 

2.3789 

.37638 

.92186 

.96467 

.2 

.9 

.38912 

.59009 

.42242 

.62574 

2.3673 

.37426 

.92119 

.96435 

.1 

23 

.39073 

.59188 

.42447 

.62785 

2.3559 

.37215 

.92050 

.96403 

67 

.1 

.39234 

.59366 

.42654 

.62996 

2.3445 

.37004 

.91982 

.96370 

.9- 

.2 

.39394 

.59543 

.42860 

.63205 

2.3332 

.36795 

.91914 

.96338 

.8 

.3 

.39555 

.59720 

.43067 

.63414 

2.3220 

.36586 

.91845 

.96305 

.7 

.4 

.39715 

.59895 

.43274 

.63623 

2.3109 

.36377 

.91775 

.96273 

.& 

.5 

.39875 

.60070 

.43481 

.63830 

2.2998 

.36170 

.91706 

.96240 

.5 

.6 

.40035 

.60244 

.43689 

.64037 

2.2889 

.35963 

.91636 

.96207 

.4 

.7 

.40195 

.60417 

.43897 

.64243 

2.2781 

.35757 

.91566 

.96174 

.3 

.8 

.40355 

.60589 

.44105 

.64449 

2.2673 

.35551 

.91496 

.96140 

.2 

.9 

.40514 

.60761 

.44314 

.64654 

2.2566 

.35346 

.91425 

.96107 

.1 

24 

.40674 

.60931 

.44523 

.64858 

2.2460 

.35142 

.91355 

.96073 

66 

.1 

.40833 

.61101 

.44732 

.65062 

2.2355 

.34938 

.91283 

.96039 

.9 

.2 

.40992 

.61270 

.44942 

.65265 

2.2251 

.34735 

.91212 

.96005 

.8 

.3 

.41151 

.61439 

.45152 

.65467 

2.2148 

.34533 

.91140 

.95971 

.r 

.4 

.41310 

.61606 

.45362 

.65669 

2.2045 

.34331 

.91068 

.95937 

.6 

.5 

.41469 

.61773 

.45573 

.65870 

2.1943 

.34130 

.90996 

.95902 

.5 

.6 

.41628 

.61939 

.45784 

.66071 

2.1842 

.33929 

.90924 

.95868 

.4 

.7 

.41787 

.62104 

.45995 

.66271 

2.1742 

.33729 

.90851 

.95833 

.3 

.8 

.41945 

.62268 

.46206 

.66470 

2.1642 

.33530 

.90778 

.95798 

.2 

.9 

.42104 

.62432 

.46418 

.66669 

2.1543 

.33331 

.90704 

.95763 

.1 

25 

.42262 

.62595 

.46631 

.66867 

2.1445 

.33133 

.90631 

.95728 

65 

Nat. 

Log. 

Nat. 

Log. 

Nat. 

Log. 

Nat. 

Loer.Ane:. 

Cosine 

Cotangent 

Tangent 

Sine 

TRIGONOMETRIC    FUNCTIONS 


Sine 

Tangent 

Cotangent 

Cosine 

Ang. 

Nat. 

Log. 

Nat. 

Log. 

Nat. 

Log. 

Nat. 

Log. 

9 

9 

10 

9 

25 

.42262 

.62595 

.46631 

.66867   2.1445 

.33133 

.90631 

.95728 

65 

.1 

.42420 

.62757 

.46843 

.67065 

.1348 

.32935 

.90557 

.95692 

.9 

.2 

.42578 

.62918 

.47056 

.67262 

.1251 

.32738 

.90483 

.95657 

.8 

3 

.42736 

.63079 

.47270 

.67458 

.1155 

.32542 

.90408 

.95621 

.7 

.4 

.42894 

.63239 

.47483 

.67654 

.1060 

.32346 

.90334 

.95585 

.6 

.5 

.43051 

.63398 

.47698 

.67850 

.0965 

.32150 

.90259 

.95549 

.5 

.6 

.43209 

.63557 

.47912 

.68044 

.0872 

.31956 

.90183 

.95513 

.4 

.7 

.43366 

.63715 

.48127 

.68239 

.0778 

.31761 

.90108 

.95476 

.3 

.8 

.43523 

.63872 

.48342 

.68432 

.0686 

.31568 

.90032 

.95440 

.2 

.9 

.43680 

.64028 

.48557 

.68626 

.0594 

.31374 

.89956 

.95403 

.1 

26 

.43837 

.64184 

.48773 

.68818 

.0503 

.31182 

.89879 

.95366 

64 

.1 

.43994 

.64339 

.48989 

.69010 

.0413 

.30990 

.89803 

.95329 

.9 

2 

.44151 

.64494 

.49206 

.69202 

.0323 

.30798 

.89726 

.95292 

.8 

'.3 

.44307 

.64647 

.49423 

.69393 

.0233 

.30607 

.89649 

.95254 

.7 

.4 

.44464 

.64800 

.49640 

.69584 

.0145 

.30416 

.89571 

.95217 

.6 

.5 

.44620 

.64953 

.49858 

.69774 

.0057 

.30226 

.89493 

.95179 

.5 

.6 

.44776 

.65104 

.50076 

.69963   1.9970 

.30037 

.89415 

.95141 

.4 

.7 

.44932 

.65255 

.50295 

.70152 

L.9883 

.29848 

.89337 

.95103 

.3 

.8 

.45088 

.65406 

.50514 

.70341   1.9797 

.29659 

.89259 

.95065 

.2 

.9 

.45243 

.65556 

.50733 

.70529   1.9711 

.29471 

.89180 

.95027 

.1 

27 

.45399 

.65705 

.50953 

.70717   1.9626 

.29283 

.89101 

.94988 

63 

.1 

.45554 

.65853 

.51173 

.70904   1.9542 

.29096 

.89021 

.94949 

.9 

.45710 

.66001 

.51393 

.71090   1.9458 

.28910 

.88942 

.94911 

.8 

!s 

.45865 

.66148 

.51614 

.71277 

1.9375 

.28723 

.88862 

.94871 

.7 

.4 

.46020 

.66295 

.51835 

.71462   1.9292 

.28538 

.88782 

.94832 

.6 

.5 

.46175 

.66441 

.52057 

.71648   1.9210 

.28352 

.88701 

.94793 

.5 

.6 

.46330 

.66586 

.52279 

.71833 

1.9128 

.28167 

.88620 

.94753 

.4 

.7 

.46484 

.66731 

.52501 

.72017 

1.9047 

.27983 

.88539 

.94714 

.3 

.8 

.46639 

.66875 

.52724 

.72201   1.8967 

.27799 

.88458 

.94674 

.2 

.9 

.46793 

.67018 

.52947 

.72384   1.8887 

.27616 

.88377 

.94634 

.1 

28 

.46947 

.67161 

.53171 

.72567 

1.8807 

.27433 

.88295 

.94593 

62 

.1 

.47101 

.67303 

.53395 

.72750 

1.8728 

.27250 

.88213 

.94553 

.9 

.2 

.47255 

.67445 

.53620 

.72932  ' 

1.8650 

.27068 

.88130 

.94513 

.8 

.3 

.47409 

.67586 

.53844 

.73114 

1.8572 

.26886 

.88048 

.94472 

.7 

.4 

.47562 

.67726 

.54070 

.73295 

1.8495 

.26704 

.87965 

.94431 

.6 

.5 

.47716 

.67863 

.54296 

.73476 

1.8418 

.26524 

.87882 

.94390 

.5 

.6 

.47869 

.68006 

.54522 

.73657 

1.8341 

.26343 

.87798 

.94349 

.4 

.7 

.48022 

.68144 

.54748 

.73837 

1.8265 

.26163 

.87715 

.94307 

.3 

.8 

.48175 

.68283 

.54975 

.74017 

1.8190 

.25983 

.87631 

.94266 

.2 

.9 

.48328 

.68420 

.55203 

.74196 

1.8115 

.25804 

.87546 

.94224 

.1 

29 

.48481 

.68557 

.55431 

.74375 

1.8040 

.25625 

.87462 

.94182 

61 

.1 

.48634 

.68694 

.55659 

.74554 

1.7966 

.25446 

.87377 

.94140 

.9 

.2 

.48786 

.68829 

.55888 

.74732 

1.7893 

.25268 

.87292 

.94098 

.8 

.3 

.48938 

.68965 

.56117 

.74910 

1.7820 

.25090 

'.87207 

.94055 

.7 

.4 

.49090 

.69100 

.56347 

.75087 

1.7747 

.24913 

.87121 

.94012 

.6 

.5 

.49242 

.69234 

.56577 

.75264 

1.7675 

.24736 

.87036 

.93970 

.5 

.6 

.49394 

.69368 

.56808 

.75441 

1.7603 

.24559 

.86949 

.93927 

.4 

.7 

.49546 

.69501 

.57039 

.75617 

1.7532 

.24383 

.86863 

.93884 

.3 

.8 

.49697 

.69633 

.57271 

.75793 

1.7461 

.24207 

.86777 

.93840 

o 

.9 

,49849 

.69765 

.57503 

.75969 

1.7391 

.24031 

.86690 

.93797 

'.1 

30 

.50000 

.69897 

.57735 

.76144 

1.7321 

.23856 

.86603 

.93753 

60 

Nat. 

Log. 

Nat. 

Log. 

Nat. 

Log. 

Nat. 

Log.Ang. 

Cosine 

Cotangent 

Tangent 

Sine 

TRIGONOMETRIC    FUNCTIONS 


Sine 

Tangent 

Cotangent 

Cosine 

Ang. 

,  Nat. 

Log. 

Nat. 

Log. 

Nat. 

Log. 

Nat. 

Log. 

9 

9 

10 

9 

30 

.50000 

.69897 

.57735 

.76144 

1.7321 

.23856 

.86603 

.93753 

60 

.1 

.50151 

.70028 

.57968 

.76319 

1.7251 

.23681 

.86515 

.93709 

.9 

2 

.50302 

.70159 

.58201 

.76493 

1.7182 

.23507 

.86427 

.93665 

.8 

.'S 

.50453 

.70288 

.58435 

.76668 

1.7113 

.23332 

.86340 

.93621 

.7 

.4 

.50603 

.70418 

.58670 

.76841 

1.7045 

.23159 

.86251 

.93577 

.6 

.5 

.50754 

.70547 

.58905 

.77015 

1.6977 

.22985 

.86163 

.93532 

.5 

.6 

.50904 

.70675 

.59140 

.77188 

1.6909 

.22812 

.86074 

.93487 

.4 

.7 

.51054 

.70803 

.59376 

.77361 

1.6842 

.22639 

.85985 

.93442 

.3 

.8 

.51204 

.70931 

.59612 

.77533 

1.6775 

.22467 

.85896 

.93397 

.2 

.9 

.51354 

.71058 

.59849 

.77706 

1.6709 

.22294 

.85806 

.93352 

.1 

31 

.51504 

.71184 

.60086 

.77877 

1.6643 

.22123 

.85717 

.93307 

59 

.1 

.51653 

.71310 

.60324 

.78049 

1.6577 

.21951 

.85627 

.93261 

.9 

.2 

.51803 

.71435 

.60562 

.78220 

1.6512 

.21780 

.85536 

.93215 

.8 

.3 

.51952 

.71560 

.60801 

.78391 

1.6447 

.21609 

.85446 

.93169 

.7 

.4 

.52101 

.71685 

.61040 

.78562 

1.6383 

.21438 

.85355 

.93123 

.6 

.5 

.52250 

.71809 

.61280 

.78732 

1.6319 

.21268 

.85264 

.93077 

.5 

.6 

.52399 

.71932 

.61520 

.78902 

1.6255 

.21098 

.85173 

.93030 

.4 

.7 

.52547 

.72055 

.61761 

.79072 

1.6191 

.20928 

.85081 

.92983 

.3 

.8 

.52696 

.72177 

.62003 

.79241 

1.6128 

.20759 

.84989 

.92936 

.2 

.9 

.52844 

.72299 

.62245 

.79410 

1.6066 

.20590 

.84897 

.92889 

.1 

32 

.52992 

.72421 

.62487 

.79579 

1.6003 

.20421 

.84805 

.92842 

58 

.1 

.53140 

.72542 

.62730 

.79747 

1.5941 

.20253 

.84712 

.92795 

.9 

.2 

.53288 

.72663 

.62973 

.79916 

1.5880 

.20084 

.84619 

.92747 

.8 

.3 

.53435 

.72783 

.63217 

.80084 

1.5818 

.19916 

.84526 

.92699 

.7 

.4 

.53583 

.72902 

.63462 

.80251 

1.5757 

.19749 

.84433 

.92651 

.6 

.5 

.53730 

.73022 

.63707 

.80419 

1.5697 

.19581 

.84339 

.92603 

.5 

.6 

.53877 

.73140 

.63953 

.80586 

1.5637 

.19414 

.84245 

.92555 

.4 

.7 

.54024 

.73259 

.64199 

.80753 

1.5577 

.19247 

.84151 

.92506 

.3 

.8 

.54171 

.73377 

.64446 

.80919 

1.5517 

.19081 

.84057 

.92457 

.2 

.9 

.54317 

.73494 

.64693 

.81086 

1.5458 

.18914 

.83962 

.92408 

.1 

33 

.54464 

.73611 

.64941 

.81252 

1.5399 

.18748 

.83867 

.92359 

57 

.1 

.54610 

.73727 

.65189 

.81418 

1.5340 

.18582 

.83772 

.92310 

.9 

.2 

.54756 

.73843 

.65438 

.81583 

1  .  5  -Sil 

.18417 

.83676 

.92260 

.8 

.3 

.54902 

.73959 

.65688 

.81748 

1.5224 

.18252 

.83581 

.92211 

.7 

.4 

.55048 

.74074 

.65938 

.81913 

1.5166 

.18087 

.83485 

.92161 

.6 

.5 

.55194 

.74189 

.66189 

.82078 

1.5108 

.17922 

.83389 

.92111 

.5 

.6 

.55339 

.74303 

.66440 

.82243 

1.5051 

.17757 

.83292 

.92060 

.4 

.7 

.55484 

.74417 

.66692 

.82407 

1.4994 

.17593 

.83195 

.92010 

.3 

.8 

.55630 

.74531 

.66944 

.82571 

1.4938 

.17429 

.83098 

.91959 

2 

.9 

.55775 

.74644 

.67197 

.82735 

1.43*2 

.17265 

.83001 

.91909 

!i 

34 

.55919 

.74756 

.67451 

.82899 

1.4826 

.17101 

.82904 

.91857 

56 

.1 

.56064 

.74868 

.67705 

.83062 

1.4770 

.16938 

.82806 

.91806 

.9 

2 

.56208 

.74980 

.67960 

.83225 

1.4715 

.16775 

.82708 

.91755 

.8 

!s 

.56353 

.75091 

.68215 

.83388 

1.4659 

.16612 

.82610 

.91703 

.4 

.56497 

.75202 

.68471 

.83551 

1.4605 

.16449 

.82511 

.91651 

!e 

.5 

.56641 

.75313 

.68728 

.83713 

1.4550 

.16287 

.82413 

.91599 

.5 

.6 

.56784 

.75423 

.68985 

.83876 

1.4496 

.16124 

.82314 

.91547 

.4 

.7 

.56928 

.75533 

.69243 

.84038 

1.4442 

.15962 

.82214 

.91495 

.3 

.8 

.57071 

.75642 

.69502 

.84200 

1.4388 

.15800 

.82115 

.91442 

.2 

.9 

.57215 

.75751 

.69761 

.84361 

1.4335 

.15639 

.82015 

.91389 

.1 

35 

.57358 

.75859 

.70021 

.84523 

1.4281 

.15477 

.81915 

.91336 

55 

Nat. 

Log. 

Nat. 

Log. 

Nat. 

Log. 

Nat. 

Losr.Ane:. 

Cosine 

Cotangent 

Tangent 

Sine 

38 


TRIGONOMETRIC    FUNCTIONS 


Sine 
Ang.  Nat.   Log. 

Tangent 
Nat.  Log. 

Cotangent 
Nat.   Log. 

Cosine 
Nat.   Log. 

9 

9 

10 

9 

35 

.57358 

.75859 

.70021 

.84523 

1 

.4281 

.15477 

.81915 

.91336 

55 

.1 

.57501 

.75967 

.70281 

.84684 

1 

.4229 

.15316 

.81815 

.91283 

.9 

.2 

.57643 

.76075 

.70542 

.84845 

1 

.4176 

.15155 

.81714 

.91230 

.8 

.3 

.57786 

.76182 

.70804 

.85006 

1 

.4124 

.14994 

.81614 

.91176 

.7 

.4 

.57928 

.76289 

.71066 

.85166 

1 

.4071 

.14834 

.81513 

.91123 

.6 

.5 

.58070 

.76395 

.71329 

.85327 

1 

.4019 

.14673 

.81412 

.91069 

.5 

.6 

.58212 

.76502 

.71593 

.85487 

1 

.3968 

.14513 

.81310 

.91014 

.4 

.7 

.58354 

.76607 

.71857 

.85647 

1 

.3916 

.14353 

.81208 

.90960 

.3 

.8 

.58496 

.76712 

.72122 

.85807 

1 

.3865 

.14193 

.81106 

.90906 

.2 

.9 

.58637 

.76817 

.72388 

.85967 

1 

.3814 

.14033 

.81004 

.90851 

.1 

36 

.58779 

.76922 

.72654 

.86126 

1 

.3764 

.13874 

.80902 

.90796 

54 

.1 

.58920 

.77026 

.72921 

.86285 

1 

.3713 

.13715 

.80799 

.90741 

.9 

2 

.59061 

.77130 

.73189 

.86445 

1 

.3663 

.13555 

.80696 

.90685 

.8 

'.3 

.59201 

.77233 

.73457 

.86604 

1 

.3613 

.13397 

.80593 

.90630 

.7 

.4 

.59342 

.77336 

.73726 

.86762 

1 

.3564 

.13238 

.80489 

.90574 

.6 

.5 

.59482 

.77439 

.73996 

.86921 

1 

.3514 

.13079 

.80386 

.90518 

.5 

.6 

.59622 

.77541 

.74267 

.87079 

1 

.3465 

.12921 

.80282 

.90462 

.4 

.7 

.59763 

.77643 

.74538 

.87238 

1 

.3416 

.12762 

.80178 

.90405 

.3 

.8 

.59902 

.77744 

.74810 

.87396 

1 

.3367 

.12604 

.80073 

.90349 

2 

.9 

.60042 

.77846 

.75082 

.87554 

1 

.3319 

.12446 

.79968 

.90292 

!i 

37 

.60182 

.77946 

.75355 

.87711 

1 

.3270 

.12289 

.79864 

.90235 

53 

.1 

.60321 

.78047 

.75629 

.87869 

1 

.3222 

.12131 

.79758 

.90178 

.9 

.2 

.60460 

.78147 

.75904 

.88027 

1 

.3175 

.11073 

.79653 

.90120 

.8 

.3 

.60599 

.78246 

.76180 

.88184 

1 

.3127 

.11816 

.79547 

.90063 

.7 

.4 

.60738 

.78346 

.76456 

.88341 

1.3079 

.11659 

.79441 

.90005 

.6 

.5 

.60876 

.78445 

.76733 

.88498 

1 

.3032 

.11502 

.79335 

.89947 

.5 

.6 

.61015 

.78543 

.77010 

.88655 

1 

.2985 

.11345 

.79229 

.89888 

.4 

.7 

.61153 

.78642 

.77289. 

.88812 

1 

.2938 

.11188 

.79122 

.89830 

.3 

.8 

.61291 

.78739 

.77568 

.88968 

1 

.2892 

.11032 

.79016 

.89771 

.2 

.9 

.61429 

.78837 

.77848 

.89125 

1 

.2846 

.10875 

.78908 

.89712 

.1 

38 

.61566 

.78934 

.78129 

.89281 

1 

.2799 

.10719 

.78801 

.89653 

52 

.1 

.61704 

.79031 

.78410 

.89437 

1 

.2753 

.10563 

.78694 

.89594 

.9 

2 

.61841 

.79128 

.78692 

.89593 

1 

.2708 

.10407 

.78586 

.89534 

.8 

'.3 

.61978 

.79224 

.78975 

.89749 

1 

.2662 

.10251 

.78478 

.89475 

.7 

.4 

.62115 

.79319 

.79259 

.89905 

1 

.2617 

.10095 

.78369 

.89415 

.6 

.5 

.62251 

.79415 

.79544 

.90061 

1 

.2572 

.09939 

.78261 

.89354 

.5 

.6 

.62388 

.79510 

.79829 

.90216 

1 

.2527 

.09784 

.78152 

.89294 

.4 

.7 

.62524 

.79605 

.80115 

.90371 

1 

.2482 

.09629 

.78043 

.89233 

.3 

.8 

.62660 

.79699 

.80402 

.90527 

1 

.2437 

.09473 

.77934 

.89173 

.2 

.9 

.62796 

.79793 

.80690 

.90682 

1 

.2393 

.09318 

.77824 

.89112 

.1 

39 

.62932 

.79887 

.80978 

.90837 

1 

.2349 

.09163 

.77715 

.89050 

51 

.1 

.63068 

.79981 

.81268 

.90992 

1 

.2305 

.09008 

.77605 

.88989 

.9 

.2 

.63203 

.80074 

.81558 

.91147 

1 

.2261 

.08853 

.77494 

.88927 

.8 

.3 

.63338 

.80166 

.81849 

.91301 

1 

.2218 

.08699 

.77384 

.88865 

.7 

.4 

.63473 

.80259 

.82141 

.91456 

1 

.2174 

.08544 

.77273 

.88803 

.6 

.5 

.63608 

.80351 

.82434 

.91610 

1 

.2131 

.08390 

.77162 

.88741 

.5 

.6 

.63742 

.80443 

.82727 

.91765 

1 

.2088 

.08235 

.77051 

.88678 

.4 

.7 

.63877 

.80534 

.83022 

.91919 

1 

.2045 

.08081 

.76940 

.88615 

.3 

.8 

.64011 

.80625 

.83317 

.92073 

1 

.2002 

.07927 

.76828 

.88552 

.2 

.9 

.64145 

.80716 

.83613 

.92227 

1 

.1960 

.07773 

.76717 

.88489 

.1 

40 

.64279 

.80807 

.83910 

.92381 

1 

.1918 

.07619 

.76604 

.88425 

51 

Nat. 

LOST. 

Nat. 

Log. 

Nat. 

Log. 

Nat. 

Log.Ane. 

Cosine  ~ 

Cotangent 

Tangent 

Sine 

TRIGONOMETRIC    FUNCTIONS 


39 


Sine 
Ang.  Nat.   Log. 

Tangent 
Nat.   Log. 

Cotangent 
Nat.   Log. 

Cosine 
Nat.   Log. 

9 

9 

10 

9 

40 

.64279 

.80807 

.83910 

.92381 

1 

.1918 

.07619 

.76604 

.88425 

50 

.1 

.64412 

.80897 

.84208 

.92535 

1.1875 

.07465 

.76492 

.88362 

.9 

2 

.64546 

.80987 

.84507 

.92689 

1 

.1833 

.07311 

.76380 

.88298 

.8 

is 

.64679 

.81076 

.84806 

.92843 

1.1792 

.07157 

.76267 

.88234 

.7 

.4 

.64812 

.81166 

.85107 

.92996 

1 

.1750 

.07004 

.76154 

.88169 

.6 

.5 

.64945 

.81254 

.85408 

.93150 

1 

.1708 

.06850 

.76041 

.88105 

.5 

.6 

.65077 

.81343 

.85710 

.93303 

1.1667 

.06697 

.75927 

.88040 

.4 

.7 

.65210 

.81431 

.86014 

.93457 

1 

.1626 

.06543 

.75813 

.87975 

.3 

.8 

.65342 

.81519 

.86318 

.93610 

1 

.1585 

.06390 

.75700 

.87909 

2 

.9 

.65474 

.81607 

.86623 

.93763 

1 

.1544 

.06237 

.75585 

.87844 

ii 

41 

.65606 

.81694 

.86929 

.93916 

1 

.1504 

.06084 

.75471 

.87778 

49 

.1 

.65738 

.81781 

.87236 

.94069 

1 

.1463 

.05931 

.75356 

.87712 

.9 

2 

.65869 

.81868 

.87543 

.94222 

1 

.1423 

.05778 

.75241 

.87646 

.8 

.3 

.66000 

.81955 

.87852 

.94375 

1 

.1383 

.05625 

.75126 

.87579 

.7 

.4 

.66131 

.82041 

.88162 

.94528 

1 

.1343 

.05472 

.75011 

.87513 

.6 

.5 

.66262 

.82126 

.88473 

.94681 

1 

.1303 

.05319 

.74896 

.87446 

.5 

.6 

.66393 

.82212 

.88784 

.94834 

1 

.1263 

.05166 

.74780 

.87378 

.4 

.7 

.66523 

.82297 

.89097 

.94986 

1 

.1224 

.05014 

.74664 

.87311 

.3 

.8 

.66653 

.82382 

.89410 

.95139 

1 

.1184 

.04861 

.74548 

.87243 

2 

.9 

.66783 

.82467 

.89725 

.95291 

1 

.1145 

.04709 

.74431 

.87175 

'.1 

42 

.66913 

.82551 

.90040 

.95444 

1 

.1106 

.04556 

.74314 

.87107 

48 

.1 

.67043 

.82635 

.90357 

.95596 

1 

.1067 

.04404 

.74198 

.87039 

.9 

.2 

.67172 

.82719 

.90674 

.95749 

1 

.1028 

.04252 

.74080 

.86970 

.8 

.3 

.67301 

.82802 

.90993 

.95901 

1 

.0990 

.04099 

.73963 

.86902 

.7 

.4 

.67430 

.82885 

.91313 

.96053 

1 

.0951 

.03947 

.73846 

.86832 

.6 

.5 

.67559 

.82968 

.91633 

.96205 

1 

.0913 

.03795 

.73728 

.86763 

.5 

.6 

.67688 

.83051 

.91955 

.96357 

1 

.0875 

.03643 

.73610 

.86694 

.4 

.7 

.67816 

.83133 

.92277 

.96510 

1 

.0837 

.03490 

.73491 

.86624 

.3 

.8 

.67944 

.83215 

.92601 

.96662 

1 

.0799 

.03338 

.73373 

.86554 

.2 

.9 

.68072 

.83297 

.92926" 

.96814 

I 

.0761 

.03186 

.73254 

.86483 

.1 

43 

.68200 

.83378 

.93252 

.96966 

1 

.0724 

.03034 

.73135 

.86413 

47 

.1 

.68327 

.83460 

.93578 

.97118 

1 

.0686 

.02882 

.73016 

.86342 

.9 

2 

.68455 

.83540 

.93906 

.97269 

1 

.0649 

.02731 

.72897 

.86271 

.8 

'.3 

.68582 

.83621 

.94235 

.97421 

1 

.0612 

.02579 

.72777 

.86200 

.7 

.4 

.68709 

.83701 

.94565 

.97573 

1 

.0575 

.02427 

.72657 

.86128 

.6 

.5 

.68835 

.83781 

.94896 

.97725 

i 

.0538 

.02275 

.72537 

.8605& 

.5 

.6 

.68962 

.83861 

.95229 

.97877 

i 

.0501 

.02123 

.72417 

.85984 

.4 

.7 

.69088 

.83940 

.95562 

.98029 

i 

.0464 

.01971 

.72297 

.85912 

.3 

.8 

.69214 

.84020 

.95897 

.98180 

i 

.0428 

.01820 

.72176 

.85839 

.2 

.9 

.  69340 

.84099 

.96232 

.98332 

i 

.0392 

.01668 

.72055 

.85766 

.1 

44 

.69466 

.84177 

.96569 

.98484 

i 

.0355 

.01516 

.71934 

.85693 

46 

.1 

.69591 

.84255 

.96907 

.98635 

i 

.0319 

.01365 

.71813 

.85620 

.9 

.2 

.69717 

.84334 

.97246 

.98787 

i 

.0283 

.01213 

.71691 

.85547 

.8 

.3 

.69842 

.84411 

.97586 

.98939 

i 

.0247 

.01061 

.71569 

.85473 

.7 

.4 

.69966 

.84489 

.97927 

.99090 

i 

.0212 

.00910 

.71447 

.85399 

.6 

.5 

.70091 

.84566 

.98270 

.99242 

i 

.0176 

.00758 

.71325 

.85324 

.5 

.6 

.70215 

.84643 

.98613 

.99394 

i 

.0141 

.00606 

.71203 

.85250 

.4 

.7 

.70339 

.84720 

.98958 

.99545 

i 

.0105 

.00455 

.71080 

.85175 

.3 

.8 

.70463 

.84796 

.99304 

.99697 

i 

.0070 

.00303 

.70957 

.85100 

.2 

.9 

.70587 

.84873 

.99652 

.99848 

i 

.0035 

.00152 

.70834 

.85024 

.1 

45 

.70711 

.84949 

1.00000 

.00000 

1.0000 

.00000 

.70711 

.84949 

45 

Nat. 

Log. 

Nat. 

Log 

Nat. 

Log. 

Nat. 

Log.Ansr. 

Cosine 

Cotangent 

Tangent 

Sine 

TRIGONOMETRIC    FUNCTIONS 
Minutes  in  Decimal  Parts  of  Degrees. 


Min.   Deg. 

1  0.01667 

2  0.03333 

3  0.05000 

4  0.06667 


0.08333 
0.10000 
0.11667 
0.13333 
0.15000 

0.16667 
0.18333 
0.20000 
0.21667 
0.23333 


Min. 

15 
16 
17 
18 
19 

20 
21 
22 
23 
24 

25 
26 
27 
28 
29 


Deg. 

0.25000 
0.26667 
0.28333 
0.30000 
0.31667 

0.33333 
0.35000 
0.36667 
0.38333 
0.40000 

0.41667 
0.43333 
0.45000 
0.46667 
0.48333 


Min.   Deg. 


30 
31 
32 
33 
34 

35 
36 
37 
38 
39 

40 
41 
42 
43 
44 


0.50000 
0.51667 
0.53333 
0.55000 
0.56667 

0.58333 
0.60000 
0.61667 
0.63333 
0.65000 

0.66667 
0.68333 
0.70000 
0.71667 
0.73333 


Min. 

45 
46 
47 
48 
49 

50 
51 
52 
53 

54 

55 
56 
57 
58 
59 


Deg. 

0.75000 
0.76667 
0.78333 
0.80000 
0.81667 

0.83333 
0.85000 
0.86667 
0.88333 
0.90000 

0.91667 
0.93337 
0.95000 
0.96667 
0.98333 


Functions  of  Certain  Angles. 


degrees         °          30          45  60          90        18°        27°      36° 


sin  — 


cos= 


tan 


-1          0 


oo  0 


ctn— 


./  3  1 


0          oo 


Angles  greater  than  90°  to  Angles  less  than  90°. 

The  following  scheme  gives,  first,  the  sign  of  the  trigono- 
metric functions  for  angles  in  the  various  quadrants,  and, 
second,  the  reduction  of  the  functions  for  angles  greater  than 
90  degrees  to  the  functions  for  angles  less  than  90  degrees. 


TRIGONOMETRIC    FUNCTIONS 
Angles  Greater  than  90°  to  Angles  less  than  90C 


41 


Angle  u  between 


Angle  u  in  degrees 


0     90    180  270 

and      and       and      and 

90    180  270  360 


=fc  u       90  ±  u    180  ±  u   270  ±  u 


sin  u  —  -f-  -j-  =fc  sin  u  -f  cos  u  -+-  sin  u  •—  cos  u 

cos  u  —  -{-  ~h  +  cos  u  =F  sin  u  —  cos  u  +  sin  u 

tan  u  =  -}-  +      —  +  tan  u  =f  ctn  u  +  tan  u  HF  ctn  u 

ctn  u  =  +  —     -f-      —  |  +  ctn  u  +  tan  u  +  ctn  u  +  tan  u 


sin  (45°  ±  u)  =  cos  (45°  =F  u)       tan  (45°  ±  u)  =  ctn  (45°  =F  u) 


Formulae. 


tan  a 


2.     cosa  = 


csc  a       i/l-fctn2a         Vl-ftanaa 

1      __       1 ctn_a 

sec  a        V7l+tan2a         l/l+ctn2a 


o  1  sin  a 

3.  tana  = 

ctn  a        cos  a 

4.  sin2  a  -f  cos2  a  =  1. 

5.  csc2  a  =  1  -f  ctn2  a.  ' 

9.  cos  yz  a  =  V] 

10.  sin  2  a  =  2  sin  a  cos  a. 


6.     sec*  a  =  1 


tan2  a. 


7.     sin  K  a--=V)£(l  —  cos  a). 


-r  cos  a).     8.     tan  %  a  =  \U    ~  cos  a 


cos  a 


11.  cos  2  a  ==  cos2  a  —  sin  a  =  1  —  ?  sin2  a  —  2  cos2  a  —  1. 

12.  sin  3  a  =  3  sin  a  —  4  sin3  a.     14.    cos  3  a  =  4  cos3  a  —  3  cos  a. 


13.     tan2a=  -_  _ 

1  —  tan2  a 


15.     ctn  2  a  = 


ctn  a 


42  TRIGONOMETRIC    FUNCTIONS 

Formulae 

16.  sin  (a  +  ft)  =  sin  a.  cos  6  +  cos  a.  sin  b. 

17.  cos  (a.  +  6)  =  cos  a.  cos  6  =F  sin  a.  sin  ft. 

18.  tan  (a  ±  ft)  =    tan  «  ±  tan  ft  . 

1  +  tan  a.  tan  ft 


19. 


ctn  6  =b  ctn  a 

20.  sin  a  ±  sin  ft  =  2  sin  %  (a  +  ft)  .cos  >£  (a  +  ft). 

21.  cos  a  +  cos  ft  =  2  cos  )£  (a  +  6)  cos  V£  (a—  ft). 

22.  cos  a    -  cos  6  =  —  2  sin  ^  (a  -|~  &)  -sin  )'g  (a  — 
23. 


cos  a.  cos  ft 
24. 


sin  a  .  sin  ft 

25.  sin2a  —  sin2ft  =  cos2ft  —  cos2a  =  sin  (a  +  6)  sin  (a  —  ft). 

26.  cos2a  —  sin2ft  =  cos2ft  —  sin2a  —  cos  (a  —  ft)  cos  (a    -  ft). 


sin  A      sin  B      sin  C 


3 


a 


sin  ^  +  sin  B  =  tan  ^  (^  +  ^)  -^        ctn- 


a—  6      sin  A  —  sin  B      tan  >     (.4  —  5)       tan  }/»  (A  —  B} 
4.     sm^A=-b(8^'  5.      cos>^^- 


6.     tanM^-= 

) 


7.     Area  =  K  be  sin  .4  =  T/S  (s  _  a)  (s  __  &)  (8  _  c). 


NOTE: — In  both  plane  and  spherical  triangles  a,  b  and  c  are  the  sides  opposite  the  angles 
A,  B  and  C  respectively. 


TRIGONOMETRIC    FUNCTIONS  43 

Plane  Oblique  Triangles 

Given        To  Find  Formula 


7,2    i    rs a2        Or,    formula    4    for 

cos  ^  =  9  7~     -    small  angles  and  5  for 

angles  near  90°. 


a,b,A          B  smB-=- 

a 

C  C  =  180°—  (A  +  B). 


a  sin  C 
sin  A 


b  cos  A  +  va2  —  b2  sin 


If  a  >  b,  then  5  <  96?°  and  B  <  A. 
If  />  >>  a  >>  &  sin  J,  then  for  one  of  the  tri- 
angles with  the  given  elements,  A  <  B  <  90° 
and  for  the  other  one  B  >  90°. 


a,  A,  B        b,  c  b  =  CT  sin  •#     r  —  «-?in_Cr  =  asin(J  +  B) 

sin  yf  sin  A  sin  ^4. 


a,  b,  C        A,  B  tan  A  =  t  a  sm  C  B^=180°—(A  +  C) 

o  —  a  cos  C 

Or,  determine  A  and  5  from 
tan  (  A~B\  =  ^—.  ctn  — ^ 


2  a  +  6  2 

c  ^V7^  +"/>2^2^aZ)  cos  (7  =  asinC  =  a-^A 

sin  ^4         cos  u 


^    F  a 
in  which  tan  -M  = 


44  TRIGONOMETRIC    FUNCTIONS 


Right-Angled  Spherical  Triangle 

NAPIER'S  RULES— The  five  circular  parts  a,  b,  (T/2-J), 
(ir/2-c),  (v/2-B),  are  supposed  to  be  ranged  around  a  circle  in 
the  order  in  which  they  naturally  occur  with  respect  to  the 
triangle.  If  any  one  of  the  parts  is  taken  for  the  middle  part, 
then  the  two  parts  next  to  it  are  the  adjacent  parts  and  the  re- 
maining two  parts  are  the  opposite  parts. 

Sine  of  middle  part  =  product  of  tangents  of  adjacent  parts. 
Sine  of  middle  part  =  product  of  cosines  of  opposite  parts. 


Any  Spherical  Triangle 


smj?  =  sjnC.  cQg  fl  =  cQg  6  cos  c  ^  sin  b  sin  c  cos  A 

sin  -a      sin  b      sin  c 

—  cos  A  =  cos  B  cos  C*  —  sin  B  sin  C  cos  a. 
sin  a  ctn  b  =  sin  C  ctn  B  +  cos  a  cos  C. 


.     1X   . /sin  (s — b}  .sin  (s — c).  !/  *        /sins. sin  (s  — 

Sin  /£  -flf — -*/  ; -       ;  COS/ 

\  sin  ^.sin  c 


. 
\      sin  ^.sm  c 


sin  8.sin(«  — a)  \         sin  B. sin 


—  cos  S. cos  (S  —  A) 
sin  £  sin  C  "\cos(S— B)  .cos(S— C) 

=  a  +  £-fc         2»9  =  ^  +  ^+C7. 


CIRCLES 


45 


CIRCLES 


Circular  Arcs,  Cords  and  Segments. 


Area  A  B  C  =  A,  Area  O  A  B  C  =&. 

The  meaning  of  L,  H,  C,  r  and  u 
are  shown  by  the  figure. 

For  a  circle  of  unit  radius,  let  /,  h,  c 
and  a  be  the  values  corresponding  to 
L,  If,  C  and  A  respectively. 


->^-  c 


L  = 


r  r  u 
180 


=--  0.01745  r  u  =  I  r. 


/  r 


//  =r  (l-cos 

A  =    **     f™  -sin  u\   = 
2     \  180  ) 


arc  1°  ^0.017453293          log  arc  1°  a±  0.241877  —  2 
arc  1'  =  0.000290888         log  arc  1'  =  0.463726  —  4 
arc  1"=  0.000004848         log  arc  1"  =  0.685575  —  6 
1  radian  =  57°  17'  44.806"  =*  57.2957795° 
I  —  number  radians  in  u  degrees 


46 


CIRCLES 


Lengths   of   Circular   Arcs   for    Unit    Radius 


DEGREES 


MINUTES        SECONDS 


U 

Length  I 

U 

Length  ' 

U 

Length  / 

U 

Length 

I  U 

Length  1 

or 

or 

or 

or 

or 

Deg.  Radians 

Deg. 

Radians 

Deg. 

Radians 

Min. 

Radians 

Sec. 

Radians 

1 

.017453 

61 

1.064651 

121 

2.111848 

1 

.000291 

1 

.0000048 

2 

.034907 

62 

1.082104 

122 

2.129302 

2 

.000582 

2 

.0000097 

3 

.052360 

63 

1.099557 

123 

2.146755 

3 

.000873 

3 

.0000145 

4 

.069813 

64 

1.117011 

124 

2.164208 

4 

.001164 

4 

.0000194 

5 

.087266 

65 

1.134464 

125 

2.181662 

5 

.001454 

5 

.0000242 

6 

.104720 

66 

1.151917 

126 

2.199115 

6 

.001745 

6 

.0000291 

7 

.122173 

67 

1.169371 

127 

2.216568 

7 

.002036 

7 

.0000339 

8 

.139626 

68 

1.186824 

128 

2.234021 

8 

.002327 

8 

.0000388 

9 

.157080 

69 

1.204277 

129 

2.251475 

9 

.002618 

9 

.0000436 

10 

.174533 

70 

1.221730 

130 

2.268928 

10 

.002909 

10 

.0000485 

11 

.191986 

71 

1.239184 

131 

2.286381 

11 

.003200 

11 

.0000533 

12 

.209440 

72 

1.256637 

132 

2.303835 

12 

.003491 

12 

.0000582 

13 

.226893 

73 

1.274090 

133 

2.321288 

13 

.003782 

13 

.0000630 

14 

.244346 

74 

1.291544 

134 

2.338741 

14 

.004072 

14 

.0000679 

15 

.261799 

75 

1.308997 

135 

2.356194 

15 

.004363 

15 

.0000727 

16 

.279253 

76 

1.326450 

136 

2.373648 

16 

.004654 

16 

.0000776 

17 

.296706 

77 

1.343904 

137 

2.391101 

17 

.004945 

17 

.0000824 

18 

.314159 

78 

1.361357 

138 

2.408554 

18 

.005236 

.18 

.0000873 

19 

.331613 

79 

1.378810 

139 

2.426008 

19 

.005527 

19 

.0000921 

20 

.349066 

80 

1.396263 

140 

2.443461 

20 

.005818 

20 

.0000970 

21 

.366519 

81 

1.413717 

141 

2.460914 

21 

.006109 

21 

.0001018 

22 

.383972 

82 

1.431170 

142 

2.478368 

22 

.006400 

22 

.0001067 

23 

.401426 

83 

1.448623 

143 

2.495821 

23 

.006690 

23 

.0001115 

24 

.418879 

84 

1.466077 

144 

2.513274 

24 

.006981 

24 

.0001164 

25 

.436332 

85 

1.483530 

145 

2.530727 

25 

.007272 

25 

.0001212 

26 

.453786 

86 

1.500983 

146 

2.548181 

26 

.007563 

26 

.0001261 

27 

.471239 

87 

1.518436 

147 

2.565634 

27 

.007854 

27 

.0001309 

28 

.488692 

88 

1.535890 

148 

2.583087 

28 

.008145 

28 

.0001357 

29 

.506145 

89 

1.553343 

149 

2.600541 

29 

.008436 

29 

.0001406 

30 

.523599 

90 

1.570796 

150 

2.617994 

30 

.008727 

30 

.0001454 

31 

.541052 

91 

1.588250 

1b1 

2.635447 

31 

.009018 

31 

.0001503 

32 

.558505 

92 

1.605703 

152 

2.652900 

32 

.009308 

32 

.0001551 

33 

.575959 

93 

1.623156 

153 

2.670354 

33 

.009599 

33 

.0001600 

34 

.593412 

94 

1.640609 

154 

2.687807 

34 

.009890 

34 

.0001648 

35 

.610865 

95 

1.658063 

155 

2.705260 

35 

.010181 

35 

.0001697 

36 

.628319 

96 

1.675516 

156 

2.722714 

36 

.010472 

36 

.0001745 

37 

.645772 

97 

1.692969 

157 

2.740167 

37 

.010763 

37 

.0001794 

38 

.663225 

98 

1.710423 

158 

2.757620 

38 

.011054 

38 

.0001842 

39 

.680678 

99 

1.727876 

159 

2.775074 

39 

.011345 

39 

.0001891 

40 

.698132 

100 

1.745329 

160 

2.792527 

40 

.011636 

40 

.0001939 

41 

.715585 

101 

1.762783 

161 

2.809980 

41 

.011926 

41 

.0001988 

42 

.733038 

102 

1.780236 

162 

2.827433 

42 

.012217 

42 

.0002U66 

43 

.750492 

103 

1.797689 

163 

2.844887 

43 

.012508 

43 

.0002085 

44 

.767945 

104 

1.815142 

164 

2.862340 

44 

.012799 

44 

.0002133 

45 

.785398 

105 

1.832596 

165 

2.879793 

45 

.013090 

45 

.0002182 

46 

.802851 

106 

1.850049 

166 

2.897247 

46 

.013381 

46 

.0002230 

47 

.820305 

107 

1.867502 

167 

2.914700 

47 

.013672 

47 

.0002279 

48 

.837758 

108 

1.884956 

168 

2.932153 

48 

.013963 

48 

.0002327 

49 

.855211 

109 

1.902409 

169 

2.949606 

49 

.014254 

49 

.0002376 

50 

.872665 

110 

1.919862 

170 

2.967060 

50 

.014544 

50 

.0002424 

51 

.890118 

111 

1.937315 

171 

2.984513 

51 

.014835 

51 

.0002473 

52 

.907571 

112 

1.954769 

172 

3.001966 

52 

.015126 

52 

.0002521 

53 

.925025 

113 

1  972222 

173 

3.019420 

53 

.015417 

53 

.0002570 

54 

.942478 

114 

1.'989675 

174 

3.036873 

54 

.015708 

54 

.0002618 

55 

.959931 

115 

2.007129 

175 

3.054326 

55 

.015999 

55 

.0002666 

56 

.977384 

116 

2.024582 

176 

3.071779 

56 

.016290 

56 

.0002715 

57 

.994838 

117 

2.042035 

177 

3.089233 

57 

.016581 

57 

.0002763 

58 

1.012291 

118 

2.059489 

178 

3.106686 

58 

.016872 

58 

.0002812 

59 

1.029744 

119 

2.076942 

179 

3.124139 

59 

.017162 

59 

.0002860 

60 

1.047198 

120 

2.094395 

180 

3.141593 

60 

.017453 

60 

.0002909 

«  CIRCLES  47 

Length  of  Cord,  Rise  and  Area  of  Circular  Segments  for  Unit  Radius 


u 

De*. 
1 
2 
3 

Cord 
c 
.0175 
.0349 
.0524 

Rise 
h 

.0000 
.0002 
.0003 

Area 
a 
.00000 
.00000 
.00001 

u 

Deg. 

46 
47 
48 

Cord 
C 

.7815 
.7975 
.8135 

Rise 

h 

.0795 
.0829 
.0865 

Area 
a 
.04176 
.04448 
.04731 

4 
5 
6 

.0698 
.0872 
.1047 

.0006 
.0010 
.0014 

.00003 
.00006 
.00010 

49 
50 
51 

.8294 
.8452 
.8610 

.0900 
.0937 
.0974 

.05025 
.05331 
.05649 

7 
8 
9 

.1221 
.1395 
.1569 

.0019 
.0024 
.0031 

.00015 
.00023 
.00032 

52 
53 

54 

.8767 
.8924 
.9080 

.1012 
.1051 
.1090 

.05978 
.06319 
.06673 

10 
11 
12 

.1743 
.1917 
.2091 

.0038 
.0046 
.0055 

.00044 
.00059 
.00076 

55 
56 
57 

.9235 
,9389 
.9543 

.1130 
.1171 
.1212 

.07039 
.07417 

.07808 

13 
14 
15 

.2264 
.2437 
.2611 

.0064 

.0075 
'.0086 

.00097 
.00121 
.00149 

58 
59 
60 

.9696 
.9848 
1.0000 

.1254 
.1296 
.1340 

.08212 
.08629 
.09059 

16 
17 
18 

.2783 
.2956 
.3129 

.0097 
.0110 
.0123 

.00181 
.00217 
.00257 

61 
62 
63 

1.0151 
1.0301 
1.0450 

.1384 
.1428 
.1474 

.09502 
.09958 
.10428 

19 
20 
21 

.3301 
.3473 
.3645 

.0137 
.0152 
.0167 

.00302 
.00352 
.00408 

64 
65 
66 

1.0598 
1.0746 
1.0893 

.1520 
.1566 
.1613 

.10911 
.11408 
.11919 

22 
23 

24 

.3816 
.3987 
.4158 

.0184 
.0201 
.0219 

.00468 
.00535 
.00607 

67 
68 
69 

1.1039 
1.1184 
1.1328 

.1661 
.1710 
.1759 

.12443 
.12982 
.13535 

25 
26 
27 

.4329 
.4499 
.4669 

.0237 
.0256 
.0276 

.00686 
.00771 
.00862 

70 
71 
72 

1.1472 
1.1614 
1.1756 

.1808 
.1859 
.1910 

.14102 
.14683 
.15279 

28 
29 
30 

.4838 
.5008 
.5176 

.0297 
.0319 
.0341 

.00961 
.01067 
.01180 

73 
74 
75 

1.1896 
.1.2036 
1.2175 

.1961 
.2014 
.2066 

.15889 
.16514 
.17154 

31 
32 
33 

.5345 
.5512 
.5680 

.0364 
.0387 
.0412 

.01301 
.01429 
.01566 

76 
77 
78 

1.2313 
1.2450 
1.2586 

.2120 
.2174 
.2229 

.17808 
.18477 
.19160 

34 
35 
36 

.5847 
.6014 
.6180 

.0437 
.0463 
.0489 

.01711 
.01864 
.02027 

79 
80 
81 

1.2722 
1.2856 
1.2989 

.2284 
.2340 
.2396 

.19859 
.20573 
.21301 

37 
38 

39 

.6346 
.6511 
.6676 

.0517 
.0545 
.0574 

.02198 
.02378 
.02568 

82 
83 
84 

1.3121 
1.3252 
1.3383 

.2453 
.2510 
.2569 

.22045 
.22804 
.23578 

40 

41 
42 

.6840 
.7004 
.7167 

.0603 
.0633 
.0664 

.02767 
.02976 
.03195 

85 
86 
87 

1.3512 
1.3640 
1.3767 

.2627 
.2686 
.2746 

.24367 
.25171 
.25990 

43 
44 

45 

.7330 
.7492 
.7654 

.0696 
.0728 
.0761 

.03425 
.03664 
.03915 

88 
89 
90 

1.3893 
1.4018 
1.4142 

.2807 
.2867 
.2929 

.26825 
.27675 
.28540 

48  CIRCLES 

Length  of  Cord,  Rise  and  Area  of  Circular  Segments  for  Unit  Radius 


u 

Deg. 
91 
92 
93 

Cord 
C 

1.4265 
1.4387 
1.4507 

Rise 
h 

.2991 
.3053 
.3116 

Area 

a 

.29420 
.30316 
.31226 

u 

Deer- 
136 
137 
138 

Cord 
C 

1.8544 
1.8608 
1.8672 

Rise 
h 

.6254 
.6335 
.6416 

Area 

a 
.83949 
.85455 
.86971 

94 
95 
96 

1.4627 
1.4746 
1.4863 

.3180 
.3244 
.3309 

.32152 
.33093 
.34050 

139 

140 
141 

1.8733 

1.8794 
1.8853 

.6498 
.6580 
.6662 

.88497 
.90034 
.91580 

97 
98 
99 

1.4979 
1.5094 
1.5208 

.3374 
.3439 
.3506 

.35021 
.36008 
.37009 

142 
143 

144 

1.8910 
1.8966 
1.9021 

.6744 

.6827 
.6910 

.93135 
.94700 
.96274 

100 
101 
102 

1.5321 
1.5432 
1.5543 

.3572 
.3639 
.3707 

.38026 
.39058 
.40104 

145 
146 
147 

1.9074 
1.9126 
1.9176 

.6993 
.7076 
.7160 

.97858 
.99449 
1.01050 

103 
104 
105 

1.5652 
1.5760 
1.5867 

.3775 
.3843 
.3912 

.41166 
.42242 
.43333 

148 

149 
150 

1.9225 
1.9273 
1.9319 

.7244 
.7328 
.7412 

1.02658 
1.04275 
1.05900 

106 
107 
108 

1.5973 
1.6077 
1.6180 

.3982 
.4052 
.4122 

.44439 
.45560 
.46695 

151 

152 
153 

1.9363 
1.9406 
1.9447 

.7496 
.7581 
.7666 

1.07532 
1.09171 
1.10818 

109 
110 
111 

1.6282 
1.6383 
1.6483 

.4193 
.4264 
.4336 

.47844 
.49008 
.50187 

154 
155 
156 

1.9487 
1.9526 
1.9563 

.7750 
.7836 
.7921 

1.12472 
1.14132 
1.15799 

112 
113 
114 

1.6581 
1.6678 
1.6773 

.4408 
.4481 
.4554 

.51379 

.52586 
.53807 

157 
158 
159 

1.9598 
1.9633 
1.9665 

.8006 
.8092 
.8178 

1.17472 
1.19151 
1.20835 

115 
116 
117 

1.6868 
1.6961 
1.7053 

.4627 
.4701 
.4775 

.55041 
.56289 
.57551 

160 
161 
162 

1.9696 
1.9726 
1.9754 

.8264 
.8350 
.8436 

1.22525 
1.24221 
1.25921 

118 
119 
120 

1.7143 
1.7233 
1.7321 

.4850 
.4925 
.5000 

.58827 
.60116 
.61418 

163 
164 
165 

1.9780 
1.9805 
1.9829 

.8522 
.8608 
.8695 

1.27626 
1.29335 
1.31049 

121 
122 
123 

1.7407 
1.7492 
1.7576 

.5076 
.5152 
.5228 

.62734 
.64063 
.65404 

166 
167 
168 

1.9851 
1.9871 
1.9890 

.8781 
.8868 
.8955 

1.32766 
1.34487 
1.36212 

124 
125 
126 

1.7659 
1.7740 
1.7820 

.5305 
.5383 
.5460 

.66759 
.68125 
.69505 

169 
170 
171 

1.9908 
1.9924 
1.9938 

.9042 
.9128 
.9215 

1.37940 
1.39671 
1.41404 

127 

128 
129 

1.7899 
1.7976 
1.8052 

.5538 
.5616 
.5695 

.70897 
.72301 
.73716 

172 
173 
174 

1.9951 
1.9963 
1.9973 

.9302 
.9390 
.9477 

1.43140 
1.44878 
1.46617 

130 
131 
132 

1.8126 
1.8199 
1.8271 

.5774 
.5853 
.5933 

.75144 
.76584 
.78034 

175 
176 
177 

1.9981 
1.9988 
1.9993 

.9564 
.9651 
.9738 

1.48359 
1.50101 
1.51845 

133 

134 
135 

1.8341 
1.8410 

1.8478 

.6013 
.6093 
.6173 

.79497 
.80970 
.82454 

178 
179 
180 

1.9997 
1.9999 
2.0000 

.9825 
.9913 
1.0000 

1.53589 
1.55334 
1.5706'J 

CIRCLES 


1!) 


Areas    and    Circumferences    of    Circles 


Dia. 

Circ. 

Area 

Dia. 

Circ. 

Area 

Dia. 

Circ. 

Area 

3-4 

2.3562 

.44179 

3 

9.4248 

7.0686 

1-64' 

.04909 

.00019 

49-64 

2.4053 

.46039 

1-16 

9.6211 

7.3662 

1-32 

.09818 

.00077 

25-32 

2.4544 

.47937 

1-8 

9.8175 

7.6699 

3-64 

.14726 

.00173 

51-64 

2.5035 

.49872 

3-16 

10.014 

7.9798 

1-16 

.19635 

.00307 

13-16 

2.5525 

.51849 

1-4 

10.210 

8.2958 

5-64 

.24545 

.00479 

53-64 

2.6017 

.53862 

5-16 

10.407 

8.6179 

3-32 

.29452 

.00690 

27-32 

2.6507 

.55914 

3-8 

10.603 

8.9462 

7-64 

.34363 

.00939 

55-64 

2.6999 

.58003 

7-16 

10.799 

9.2806 

1-8 

.39270 

.01227 

7-8 

2.7489 

.60132 

1-2 

10.996 

9.6211 

9-64 

.44181 

.01553 

57-64 

2.7981 

.62298 

9-16 

11.192 

9.9678 

5-32 

.49087 

.01917 

29-32 

2.8471 

.64504 

5-8 

11.388 

10.321 

11-64 

.53999 

.02320 

59-64 

2.8962 

.66746 

11-16 

11.585 

10.680 

3-16 

.58905 

.02761 

15-16 

2.9452 

.69029 

3-4 

11.781 

11.045 

13-64 

.63817 

.03241 

61-64 

2.9944 

.71349 

13-16 

11.977 

11.416 

7-32 

.68722 

.03758 

31-32 

3.0434 

.73708 

7-8 

12.174 

11.793 

15-64 

.73635 

.04314 

63-64 

3.0925 

,76097 

15-16 

12.370 

12.177 

1-4 

.78540 

.04909 

1 

3.1416 

.7854 

4 

12.566 

12.566 

17-64 

.83453 

.05542 

1-16 

3.3379 

.8866 

1-16 

12.763 

12.962 

9-32 

.88357 

.06213 

1-8 

3.5343 

.9940 

1-8 

12.959 

13.364 

19-64 

.93271 

.06922 

3-16 

3.7306 

1.1075 

3-16 

13.155 

13.772 

5-16 

.98175 

.07670 

1-4 

3.9270 

1.2272 

1-4 

13.352 

14.186 

21-64 

1.0309 

.08456 

5-16 

4.1233 

1.3530 

5-16 

13.548 

14.607 

11-32 

1.0799 

.09281 

3-8 

4.3197 

1.4849 

3-8 

13.744 

15.033 

23-64 

1.1291 

.10144 

7-16 

4.5160 

1.6230 

7-16 

13.941 

15.466 

3-8 

1.1781 

.11045 

1-2 

4.7124 

1.7671 

1-2 

14.137 

15.904 

25-64 

1.2273 

.11984 

9-16 

4.9087 

1.9175 

9-16 

14.334 

16.349 

13-32 

1.2763 

.12962 

5-8 

5.1051 

2.0739 

5-8 

14.530 

16.800 

27-64 

1.3254 

.13979 

11-16 

5.3014 

2.2365 

11-16 

14.726 

17.257 

7-16 

1.3744 

.15033 

3-4 

5.4978 

2.4053 

3-4 

14.923 

17.721 

29-64 

1.4236 

.16126 

13-16 

5.6941 

2.5802 

13-16 

15.119 

18.190 

15-32 

1.4726 

.17257 

7-8 

5.8905 

2.7612 

7-8 

15.315 

18.665 

31-64 

1.5218 

.18427 

15-16 

6.0868 

2.9483 

'15-16 

15.512 

19.147 

1-2 

1.5708 

.19635 

2 

6.2832 

3.1416 

5 

15.708 

19.635 

33-64 

1.6199 

.20880 

1-16 

6.4795 

3.3410 

1-16 

15.904 

20.129 

17-32 

1.6690 

.22166 

1-8 

6.6759 

3.5466 

1-8 

16.101 

20.629 

35-64 

1.7181 

.23489 

3-16 

6.8722 

3.7583 

3-16 

16.297 

21.135 

9-16 

1.7671 

.24850 

1-4 

7.0686 

3.9761 

1-4 

14.493 

21.648 

37-64 

1.8163 

.26248 

5-16 

7.2649 

"4.2000 

5-16 

16.690 

22.166 

19-32 

1.8653 

.27688 

3-8 

7.4613 

4.4301 

3-8 

16.886 

22.691 

39-64 

1.9144 

.29164 

7-16 

7.6576 

4.6664 

7-16 

17.082 

23.221 

5-8 

1.9635 

.30680 

1-2 

7.8540 

4.9087 

1-2 

17.279 

23.758 

41-64 

2.0126 

.32232 

9-16 

8.0503 

5.1572 

9-16 

17.475 

24.301 

21-32 

2.0617 

.33824 

5-8 

8.2467 

5.4119 

5-8 

17.671 

24.850 

43-64 

2.1108 

.35453 

11-16 

8.4430 

5.6727 

11-16 

17.868 

25.406 

11-16 

2.1598 

.37122 

3-4 

8.6394 

5.9396 

3-4 

18.064 

25.967 

45-64 

2.2090 

.38828 

13-16 

8.8357 

6.2126 

13-16 

18.261 

26.535 

23-32 

2.2580 

.40574 

7-8 

9.0321 

6.4918 

7-8 

18.457 

27.109 

47-64 

2.3071 

.42356 

15-16 

9.2284 

6.7771 

15-16 

18.653 

27.688 

CIRCLES 
Areas  and   Circumferences   of  Circles 


Dia. 

Circ. 

Area 

Dia. 

Circ. 

Area 

Dia. 

Circ. 

Area 

6 

18.850 

28.274 

13 

40.841 

132.73 

20 

62.832 

314.16 

1-8 

19.242 

29.465 

1-8 

41.233 

135.30 

1-8 

63.225 

318.10 

1-4 

19.635 

30.680 

1-4 

41.626 

137.89 

1-4 

63.617 

322.06 

3-8 

20.028 

31.919 

3-8 

42.019 

140.50 

3-8 

64.010 

326.05 

1-2 

20.420 

33.183 

1-2 

42.412 

143.14 

1-2 

64.403 

330.06 

5-8 

20.813 

34.472 

5-8 

42.804 

145.80 

5-8 

64.795 

334.10 

3-4 

21.206 

35.785 

3-4 

43.197 

148.49 

3-4 

65.188 

338.16 

7-8 

21.598 

37.122 

7-8 

43.590 

151.20 

7-8 

65.581 

342.25 

7 

21.991 

38.485 

14 

43.982 

153.94 

21 

65.973 

346.36 

1-8 

22.384 

39.871 

1-8 

44.375 

156.70 

1-8 

66.366 

350.50 

1-4 

22.776 

41.282 

1-4 

44.768 

159.48 

1-4 

66.759 

354.66 

3-8 

23.169 

42.718 

3-8 

45.160 

162.30 

3-8 

67.152 

358.84 

1-2 

23.562 

44.179 

1-2 

45.553 

165.13 

1-2 

67.544 

363.05 

5-8 

23.955 

45.664 

5-8 

45.946 

167.99 

5-8 

67.937 

367.28 

3-4 

24.347 

47.173 

3-4 

46.338 

170.87 

3-4 

68.330 

371.54 

7-8 

24.740 

48.707 

7-8 

46.731 

173.78 

7-8 

68.722 

375.83 

8 

25.133 

50.265 

15 

47.124 

176.71 

22 

69.115 

380.13 

1-8 

25.525 

51.849 

1-8 

47.517 

179.67 

1-8 

69.508 

384.46 

1-4 

25.918 

53.456 

1-4 

47.909 

182.65 

1-4 

69.900 

388.82 

3-8 

26.311 

55.088 

3-8 

48.302 

185.66 

3-8 

70.293 

393.20 

1-2 

26.704 

56.745 

1-2 

48.695 

188.69 

1-2 

70.686 

397.61 

5-8 

27.096 

58.426 

5-8 

49.087 

191.75 

5-8 

71.079 

402.04 

3-4 

27.489 

60.132 

3-4 

49.480 

194.83 

3-4 

71.471 

406.49 

7-8 

27.882 

61.862 

7-8 

49.873 

197.93 

7-8 

71.864 

410.97 

9 

1-8 
1-4 
3-8 
1-2 

5-8 
3-4 

7-8 

28.274 
28.667 
29.060 
29.452 
29.845 
30.238 
30.631 
31.023 

63.617 
65.397 
67.201 
69.029 
70.882 
72.760 
74.062 
76.589 

16 
1-8 
1-4 
3-8 
1-2 
5-8 
3-4 
7-8 

50.265 
50.658 
51.051 
51.444 
51.836 
52.229 
52.622 
53.014 

201.06 
204.22 
207.39 
210.60 
213.82 
217.08 
220.35 
223.65 

23 

1-8 
1-4 
3-8 
1-2 
5-8 
3-4 
7-8 

72.257 
72.649 
73.042 
73.435 
73.827 
74.220 
74.613 
75.006 

415.48 
420.00 
424.56 
429.13 
433.74 
438.36 
443.01 
447.69 

10 
1-8 
1-4 
3-8 
1-2 
5-8 
3-4 

31.416 
31.809 
32.201 
32.594 
32.987 
33.379 
33.772 

78.540 
80.516 
82.516 
84.541 
86.590 
88.664 
90.763 

17 
1-8 
1-4 
3-8 
1-2 
5-8 
3-4 

53.407 
53.800 
54.192 
54.585 
54.978 
55.371 
55.763 

226.98 
230.33 
233.71 
237.10 
240.53 
243.98 
247.45 

24 
1-8 
1-4 
3-8 
1-2 
6-8 
3-4 

75.398 
75.791 
76.184 
76.576 
76.969 
77.362 
77.754 

452.39 
457.11 
461.86 
466.64 
471.44 
476.26 
481.11 

7-8 

34.165 

92.886 

7-8 

56.156 

250.95 

7-8 

78.147 

485.98 

11 

34.558 

95  033 

18 

56.549 

254.47 

25 

78.540 

490.87 

1-8 

34.950 

97.205 

1-8 

56.941 

258.02 

1-8 

78.933 

495.79 

1-4 

35.343 

99.402 

1-4 

57.334 

261.59 

1-4 

79.325 

500.74 

3-8 

35.736 

101.62 

3-8 

57.727 

265.18 

3-8 

79.718 

505.71 

1-2 

36.128 

103.87 

1-2 

58.119 

268.80 

1-2 

80.111 

510.71 

5-8 

36.521 

106.14 

5-8 

58.512 

272.45 

5-8 

80.503 

515.72 

3-4 

36.914 

108.43 

3-4 

58.905 

276.12 

3-4 

80.896 

520.77 

7-8 

37.306 

110.75 

7-8 

59.298 

279.81 

7-8 

81.289 

525.84 

12 

37.699 

113.10 

19 

59.690 

283.53 

26 

81.681 

530.93 

1-8 

38.092 

115.47 

1-8 

60.083 

287.27 

1-8 

82.074 

536.05 

1-4 

38.485 

117.86 

1-4 

60.476 

291.04 

1-4 

82.467 

541.19 

3-8 

38.877 

120.28 

3-8 

60.868 

294.83 

3-8 

82.860 

546.35 

1-2 

39.270 

122.72 

1-2 

61.261 

298.65 

1-2 

83.252 

551.55 

5-8 

39.663 

125.19 

5-8 

61.654 

302.49 

5-8 

83.645 

556.76 

3-4 

40.055 

127.68 

3-4 

62.046 

306.35 

3-4 

84.038 

562.00 

7-8 

40.448 

130.19 

7-8 

62.439 

310.24 

7-8 

84.430 

567.27 

CIRCLES 
Areas  and   Circumferences   of  Circles 


51 


Dia. 

Circ. 

Area 

Dia. 

Circ. 

Area 

Dia. 

Circ. 

Area 

27 

84.823 

572.56 

30 

94.248 

706.86 

33 

103.673 

855.30 

1-8 

85.216 

577.87 

1-8 

94.640 

712.76 

1-8 

104.065 

861.7D 

1-4 

85.608 

583.21 

1-4 

95.033 

718.69 

1-4 

104.458 

868.31 

3-8 

86.001 

588.57 

3-8 

95.426 

724.64 

3-8 

104.851 

874.85 

1-2 

86.394 

593.96 

1-2 

95.819 

730.62 

1-2 

105.243 

881.41 

5-8 

86.786 

599.37 

5-8 

96.211 

736.62 

5-8 

105.636 

888.00 

3-4 

87.179 

604.81 

3-4 

96.604 

742.64 

3-4 

106.029 

894.62 

7-8 

87.572 

610.27 

7-8 

96.997 

748.69 

7-8 

106.421 

901.26 

28 

87.965 

615.75 

31 

97.389 

754.77 

34 

106.814 

907.92 

1-8 

88.357 

621.26 

1-8 

97.782 

760.87 

1-8 

107.207 

914.61 

1-4 

88.750 

626.80 

1-4 

98.175 

766.99 

1-4 

107.600 

921.32 

3-8 

89.143 

632.36 

3-8 

98.567 

773.14 

3-8 

107.992 

928.06 

1-2 

89.535 

637.94 

1-2 

98.960 

779.31 

1-2 

108.385 

934.82 

5-8 

89.928 

643.55 

5-8 

99.353 

785.51 

5-8 

108.778 

941.61 

3-4 

90.321 

649.18 

3-4 

99.746 

791.73 

3-4 

109.170 

948.42 

7-8 

90.713 

654.84 

7-8 

100.138 

797.98 

7-8 

109.563 

955.25 

29 

91.106 

660.52 

32 

100.531 

804.25 

35 

109.956 

962.11 

1-8 

91.499 

666.23 

1-8 

100.924 

810.54 

1-8 

110.348 

969.00 

1-4 

91.892 

671.96 

1-4 

101.316 

816.86 

1-4 

110.741 

975.91 

3-8 

92.284 

677.71 

3-8 

101.709 

823.21 

3-8 

111.134 

982.84 

1-2 

92.677 

683.49 

1-2 

102.102 

829.58 

1-2 

111.527 

989.80 

5-8 

93.070 

689.30 

5-8 

102.494 

835.97 

5-8 

111.919 

996.78 

3-4 

93.462 

695.13 

3-4 

102.887 

842.39 

3-4 

112.312 

1003.8 

7-8 

93.855 

700.98 

7-8 

103.280 

848.83 

7-8 

112.705 

1010.8 

Areas  of  Circles 


Dia.  Area      Area 
in       in            in 
in.  sq.  in.    sq.  ft- 

Dia.  Area    Area 
in        in          in 
in    sq.  in-  sq.  ft. 

Dia.  Area     Area 
in        in           in 
in  sq.  in.      sq.  ft. 

Dia.     Area     Area 
in           in          in 
in.        sq.  in.  sq.  ft 

1/4.     .0491  .00034 
%     .1104  .00077 
Vz     .1963  .00136 
3A     .4418  .00307 

10  78.540  .54542 
11   95.033  .65995 
12  113.10  .78540 
13  132.73  .92174 

27  572.56  3.9761 
28  615.75  4.2760 
29  660.52  4.5870 
30  706.86  4.9087 

44  1520.5  10.560 
45  1590.4  11.045 
46  1661.9  11.541 
47  1734.9  12.048 

1         .7854  .00545 
1'/4  1.2272  .00852 
1'/2  1.7671  .01227 
2       3.1416  .02182 

14  153.94  1.0690 
15  176.71  1.2272 
16  201.06  1.3963 
17  226.98  1.5762 

31  754.77  5.2415 
32  804.25  5.5851 
33  855.30  5.9396 
34  907.92  6.3050 

48  1809.6  12.56$ 
49  1885.7  13.095 
50  1963.5  13.635 
51   2042.S  14.186 

21/2  4.9087  .03409 
3       7.0686  .04909 
31/2  9.6211  .06681 
4       12.566  .08726 

18  254.47  1.7671 
19  283.53  1.9689 
20  314.16  2.1817 
21  346.36  2.4053 

35  962.11  6.6813 
36  1017.9  7.0686 
37  1075.2  7.4667 
38  1134.1  7.8757 

52  2123.7  14.748 
53  2206.2  15.321 
54  2290.2  15.904 
55  2375.8  16.499- 

5       19.635  .13635 
6       28.274  .19635 
7       38.484  .26725 
8       50.265  .34906 
9       63.617  .44178 

22  380.13  2.6398 
23  415.48  2.8853 
24  452.39  3.1416 
25  490.87  3.4088 
26  530.93  3.6870 

39  1194.6  8.2958 
40  1256.6  8.7265 
41   1320.2  9.1681 
42  1385.4  9.6208 
43  1452.2  10.085 

56  2463.0  17.104 
57  2551.8  17.721 
52  2642.1  18.348 
59  2734.0  18.986 
60  2827.4  19.635- 

FUNCTIONS     OF    THE    NATURAL    NUMBERS 


No. 

Square 

Cube 

Square 
Root 

Cube 
Root 

1000  x 
Recip. 

No.—  Dia. 

Circum. 

Area 

1 

2 

3 

1 
4 
9 

1 
8 
27 

1.0000 
1.4142 
1.7321 

1.0000 
1.2599 
1.4422 

1000.000 
500.000 
333.333 

3.142 

6.283 
9.425 

0.7854 
3.1416 

7.0686 

4 
5 
6 

16 
25 
36 

64 
125 
216 

2.0000 
2.2361 
2.4495 

1.5874 
1.7100 
1.8171 

250.000 
200.000 
166.667 

12.566 
15.708 
18.850 

12.5664 
19.6350 
28.2743 

7 
8 
9 

49 
64 
81 

343 

512 

729 

2.6458 
2.8284 
3.0000 

1.9129 
2.0000 
2.0801 

142.857 
125.000 
111.111 

21.991 
25.133 

28.274 

38.4845 
50.2655 
63.6173 

10 
31 

12 
13 

100 
121 
144 
169 

1000 
1331 
1728 
2197 

3.1623 
3.3166 
3.4641 
3.6056 

2.1544 
2.2240 
2.2894 
2.3513 

100.000 
90.9091 
83.3333 
76.9231 

31.416 
34.558 
37.699 
40.841 

78.5398 
95.0332 
113.097 
132.732 

14 
15 
16 

196 

225 
256 

2744 
3375 
4096 

3.7417 
3.8730 
4.0000 

2.4101 
2.4662 
2.5198 

71.4286 
66.6667 
62.5000 

43.982 
47.124 
50.265 

153.938 
176.715 

201.062 

17 
18 
19 

289 
324 
361 

4913 
5832 
6859 

4.1231 
4.2426 
4.3589 

2.5713 

2.6207 
2.6684 

58.8235 
55.5556 
52.6316 

53.407 
56.549 
59.690 

226.980 
254.469 
283.529 

20 
21 

22 
23 

400 
441 
484 
529 

8000 
9261 
10648 
12167 

4.4721 
4.5826 
4.6904 
4.7958 

2.7144 
2.7589 
2.8020 
2.8439 

50.0000 
47.6190 
45.4545 
43.4783 

62.832 
65.973 
69.115 
72.257 

314.159 
346.361 
380.133 
415.476 

24 
25 
26 

576 

625 
676 

13824 
15625 

17576 

4.8990 
5.0000 
5.0990 

2.8845 
2.9240 
2.9625 

41.6667 
40.0000 
38.4615 

75.398 

78.540 
81.681 

452.389 
490.874 
530.929 

27 
28 
29 

729 

784 
841 

19683 
21952 
24389 

5.1962 
5.2915 
5.3852 

3.0000 
3.0366 
3.0723 

37.0370 
35.7143 

34.4828 

84.823 
87.965 
91.106 

572.555 
615.752 
660.520 

30 
31 

32 
33 

900 
961 
1024 
1089 

27000 
29791 
32768 
35937 

5.4772 
5.5678 
5.6569 
5.7446 

3.1072 
3.1414 
3.1748 
3  .  2075 

33.3333 
32.2581 
31.2500 
30.3030 

94.248 
97.389 
100.531 
103.673 

706.858 
754.768 
804.248 
855.299 

34 
35 

36 

1156 
1225 
1296 

39304 
42875 
46656 

5.8310 
5.9161 
6.0000 

3.2396 
3.2711 
3.3019 

29.4118 
28.5714 

27.7778 

106.814 
109.956 
113.097 

907.920 
962.113 
1017.88 

37 
38 
39 

1369 
1444 
1521 

50653 
54872 
59319 

6.0828 
6J644 
6.2450 

3.3322 
3.3620 
3.3912 

27.0270 
26.3158 
25.6410 

116.239 
119.381 
122.522 

1075.21 
1134.11 
1194.59 

40 
41 
42 
43 

1600 
1681 
1764 
1849 

64000 
68921 
74088 
79507 

6.3246 
6.4031 
6.4807 
6.5574 

3.4200 
3.4482 
3.4760 
3.5034 

25.0000 
24.3902 
23.8095 
23.2558 

125.66 
128.81 
131.95 
135.09 

1256.64 
1320.25 
1385.44 
1452.20 

44 
45 
46 

1936 
2025 
2116 

85184 
91125 
97336 

6.6332 

6.7082 
6.7823 

3.5303 
3.5569 
3.5830 

22.7273 
22.2222 
21.7391 

138.23 
141.37 
144.51 

3520.53 
1590.43 
1661.90 

47 
48 
49 

2209 
2304 
2401 

103823 
110592 
117649 

6.8557 
6.  9282 
7  .  0000 

3.6088 
3.6342 
3.6593 

21.2766 
20.8333 

20.4082 

147.65 
150.80 
153.94 

1734.94 

1809.56 
1885.74 

FUNCTIONS    OF    THE    NATURAL    NUMBERS 


5:: 


No. 

Square 

Cube 

Square 
Root 

Cube 
Root 

1000  x 
Recip. 

No.=Dia. 

Circum. 

Area 

50 
51 
52 
53 

2500 
2601 
2704 
2809 

125000 
132651 
140608 

148877 

7.0711 
7.1414 
7.2111 
7.2801 

3.6840 
3.7084 
3.7325 
3.7563 

20.0000 
19.6078 
19.2308 
18.8679 

157.08 
160.22 
163.36 
166.50 

1963.50 
2042.82 
2123.72 
2206.18 

54 
55 
56 

2916 
3025 
3136 

157464 
166375 
175616 

7.3485 
7.4162 
7.4833 

3.7798 
3.8030 
3.8259 

18.5185 
18.1818 
17.8571 

169.65 
172.79 
175.93 

2290.22 
2375.83 
2463.01 

57 
58 
59 

3249 
3364 
3481 

185193 
195112 
205379 

7.5498 
7.6158 
7.6811 

3.8485 
3.8709 
3.8930 

17.5439 
17.2414 
16.9492 

179.07 
182.21 
185.35 

2551.76 

2642.08 
2733.97 

60 
61 
62 
63 

3600 
3721 
3844 
3969 

216000 
226981 
238328 
250047 

7.7460 
7.8102 
7.8740 
7.9373 

3.9149 
3.9365 
3.9579 
3.9791 

16.6667 
16.3934 
16.1290 
15.8730 

188.50 
191.64 
194.78 
197.92 

2827.43 
2922.47 
3019.07 
3117.25 

64 
65 

66 

4096 
4225 
4356 

262144 
274625 
287496 

8.0000 
8.0623 
8.1240 

4.0000 
4.0207 
4.0412 

15.6250 
15.3846 
15.1515 

201.06 
204.20 
207.35 

3216.99 
3318.31 
3421.19 

67 
68 
69 

4489 
4624 
4761 

300763 
314432 
328509 

8.1854 
8.2462 
8.3066 

4.0615 
4.0817 
4.1016 

14.9254 
14.7059 
14.4928 

210.49 
213.63 
216.77 

3525.65 
3631.68 
3739.28 

70 
71 

72 
73 

4900 
5041 
5184 
5329 

343000 
357911 
373248 
389017 

8.3666 
8.4261 
8.4853 
8.5440 

4.1213 
4.1408 
4.1602 
4.1793 

14.2857 
14.0845 
13.8.889 
13.6986 

219.91 
223.05 
226.19 
229.34 

3848.45 
3959.19 
4071.50 
4185.39 

74 
75 

76 

5476 
5625 
5776 

405224 
421875 
438976 

8.6023 
8.6603 

8.7178 

4.1983 
4.2172 
4.2358 

13.5135 
13.3333 
13.1579 

232.48 
235.62 
238.76 

4300.84 
4417.86 
4536.46 

77 
78 
79 

5929 
6084 
6241 

456533 
474552 
493039 

8.7750 
8.8318 
8.8882 

4  .  2543 

4.2727 
4.2908 

12.9870 
12.8205 
12.6582 

241.90 
245.04 
248.19 

4656.63 
4778.36 
4901.67 

80 

81 
82 
83 

6400 
6561 
6724 

6889 

512000 
531441 
551368 
571787 

8.9443 
9.0000 
9.0554 
9.1104 

4.3089 
4.3267 
4.3445 
4.3621 

12.5000 
12.3457 
12.1951 
12.0482 

251.33 

254.47 
257.61 
260.75 

5026.55 
5153.00 
5281.02 
5410.61 

84 
85 

86 

7056 
7225 
7396 

592704 
614125 
636056 

9.1652 
9.2195 
9.2736 

4.3795 
4.3968 
4.4140 

11.9048 
11.7647 
11.6279 

263.89 
267.04 
270.18 

5541.77 
5674.50 

5808.80 

87 
88 
89 

7569 
7744 
7921 

658503 
681472 
704969 

9.3274 
9.3808 
9.4340 

4.4310 
4.4480 
4.4647 

11.4943 
11.3636 
11.2360 

273.32 

276.46 
279.60 

5944.68 
6082.12 
6221.14 

90 

91 
92 
93 

8100 
8281 
8464 
8649 

729000 
753571 
778688 
804357 

9.4868 
9.5394 
9.5917 
9.6437 

4.4814 
4.4979 
4.5144 
4.5307 

11.1111 

10.9890 
10.8696 
10.7527 

282.74 
285.88 
289.03 
292.17 

6361.73 
6503.88 
6647.61 
6792.91 

94 
95 

96 

8836 
9025 
9216 

830584 
857375 
884736 

9.6954 
9.7468 
9.7980 

4.5468 
4.5629 
4.5789 

10.6383 
10.5263 
10.4167 

295.31 
298.45 
301.59 

6939.78 

7088.22 
7238.23 

97 
98 
99 

9409 
9604 
9801 

912673 
941192 
970299 

9.8489 
9.8995 
9.9499 

4.5947 
4.6104 
4.6261 

10.3093 
10.2041 
10.1010 

304.73 

307.88 
311.02 

7389.81 
7542.96 
7697.69 

54 


FUNCTIONS    OF    THE    NATURAL    NUMBERS 


No. 

Square 

Cube 

Square 
Root 

Cube 
Root 

1000  x 
Recip. 

No.=Dia. 
Circum.   Area 

100 
101 
102 
103 

10000 
10201 
10404 
10609 

1000000 
1030301 
1061208 
1092727 

10.0000 
10.0499 
10.0995 
10.1489 

4.6416 

4.6570 
4.6723 
4.6875 

10.0000 
9.90099 
9.80392 
9.70874 

314.16 
317.30 
320.44 
323.58 

7853.98 
8011.85 
8171.28 
8332.29 

104 
105 
'106 

10816 
11025 
11236 

1124864 
1157625 
1191016 

10.1980 
10.2470 
10.2956 

4.7027 
4.7177 
4.7326 

9.61538 
9.52381 
9.43396 

326.73 
329.87 
333.01 

8494.87 
8659.01 
8824.73 

107 
108 
109 

11449 
11664 
11881 

1225043 
1259712 
1295029 

10.3441 
10.3923 
10.4403 

4.7475 
4.7622 
4.7769 

9.34579 
9.25926 
9.17431 

336.15 
339.29 
342.43 

8992.02 
9160.88 
9331.32 

110 
111 
112 
113 

12100 
12321 
12544 
12769 

1331000 
1367631 
1404928 
1442897 

10.4881 
10.5357 
10.5830 
10.6301 

4.7914 

4.8059 
4.8203 
4.8346 

9.09091 
9.00901 

8.92857 
8.84956 

345.58 
348.72 
351.86 
355.00 

9503.32 
9676.89 
9852.03 
10028.7 

114 
115 
116 

12996 
13225 
13456 

1481544 
1520875 
1560896 

10.6771 
10.7238 
10.7703 

4.8488 
4.8629 
4.8770 

8.77193 
8.69565 
8.62069 

358.14 
361.28 
364.42 

10207.0 
10386.9 
10568.3 

117 
118 
119 

13689 
13924 
14161 

1601613 
1643032 
1685159 

10.8167 
10.8628 
10.9087 

4.8910 
4.9049 
4.9187 

8.54701 
8.47458 
8.40336 

367.57 
370.71 
373.85 

10751.3 
10935.9 
11122.0 

120 

121 
122 
123 

14400 
14641 

14884 
15129 

1728000 
1771561 
1815848 
1860867 

10.9545 
11.0000 
11.0454 
11.0905 

4.9324 
4.9461 
4.9597 
4.9732 

8.33333 
8.26446 
8.19672 
8.13008 

376.99 
380.13 
383.27 
386.42 

11309.7 
11499.0 
11689.9 
11882.3 

124 
125 
126 

15376 
15625 
15876 

1906624 
1953125 
2000376 

11.1355 
11.1803 
11.2250 

4.9866 
5.0000 
5.0133 

8.06452 
8.00000 
7.93651 

389.56 
392.70 
395.84 

12076.3 
12271.8 
12469.0 

127 
128 
129 

16129 
16384 
16641 

2048383 
2097152 
2146689 

11.2694 
11.3137 
11.3578 

5.0265 
5.0397 
5.0528 

7.87402 
7.81250 
7.75194 

398.98 
402.12 
405.27 

12667.7 
12868.0 
13069.8 

130 

131 
132 
133 

16900 
17161 
17424 
17689 

2197000 
2248091 
2299968 
2352637 

11.4018 
11.4455 
11.4891 
11.5326 

5.0658 
5.0788 
5.0916 
5.1045 

7.69231 
7.63359 
7.57576 

7.51880 

408.41 
411.55 
414.69 
417.83 

13273.2 

13478.2 
13684.8 
13892.9 

134 
135 
136 

17956 
18225 
18496 

2406104 
2460375 
2515456 

11.5758 
11.6190 
11.6619 

5.1172 
5.1299 
5.1426 

7.46269 
7.40741 
7.35294 

420.97 
424.12 
427.26 

14102.6 
14313.9 
14526.7 

137 
138 
139 

18769 
19044 
19321 

2571353 
2628072 
2685619 

11.7047 
11.7473 
11.7898 

5.1551 
5.1676 
5.1801 

7.29927 
7.24638 
7.19424 

430.40 
433.54 
436.68 

14741.1 
14957.1 
15174.7 

140 
141 
142 
143 

19600 
19881 
20164 
20449 

2744000 
2803221 
2863288 
2924207 

11.8322 
11.8743 
11.9164 
11.9583 

5.1925 

5.2048 
5.2171 
5.2293 

7.14286 
7.09220 
7.04255 
6.99301 

439.82 
442.96 
446.11 
449.25 

15393.8 
15614.5 
15836.8 
16060.6 

144 
145 
146 

20736 
21025 
21316 

2985984 
3048625 
3112136 

12.0000 
12.0416 
12.0830 

5.2415 
5.2536 
5.2656 

6.94444 
6.89655 

6.84932 

452.39 
455.53 

458.67 

16286.0 
16513.0 
16741.5 

147 
148 
149 

21609 
21904 
22201 

3176523 
3241792 
3307949 

12.1244 
12.1655 
12.2066 

5.2776 
5.2896 
5.3015 

6.80272 
6.75676 
6.71141 

461.81 
464.96 
468.10 

16971.7 
17203.4 
17436.6 

FUNCTIONS    OF    THE    NATURAL    NUMBERS 


No. 

Square 

Cube 

Square 
Root 

Cube 
Root 

1000  x 
Recip. 

No.—  Dia. 

Circum.   Area 

150 

151 
152 
153 

22500 
22801 
23104 
23409 

3375000 
3442951 
3511808 
3581577 

12.2474 

12.2882 
12.3288 
12.3693 

5.3133 
5.3251 
5.3368 
5.3485 

6.66667 
6.62252 
6.57895 
6.53595 

471.24 

474.38 
477.52 
480.66 

17671.5 
17907.9 
18145.8 
18385.4 

154 
155 
156 

23716 
24025 
24336 

3652264 
3723875 
3796416 

12.4097 
12.4499 
12.4900 

5.3601 
5.3717 
5.3832 

6.49351 
6.45161 
6.41026 

483.81 
486.95 
490.09 

18626.5 
18869.2 
19113.4 

157 
158 
159 

24649 
24964 
25281 

3869893 
3944312 
4019679 

12.5300 
12.5698 
12.6095 

5.3947 
5.4061 
5.4175 

6.36943 
6.32911 
6.28931 

493.23 
496.37 
499.51 

19359.3 
19606.7 
19855.7 

160 

161 
162 
163 

25600 
25921 
26244 
26569 

4096000 
4173281 
4251528 
4330747 

12.6491 
12.6886 
12.7279 
12.7671 

5.4288 
5.4401 
5.4514 
5.4626 

6.25000 
6.21118 
6.17284 
6.13497 

502.65 
505.80 
508.94 
512.08 

20106.2 
20358.3 
20612.0 

20867.2 

164 
165 
166 

26896 
27225 
27556 

4410944 
4492125 
4574296 

12.8062 
12.8452 
12.8841 

5.4737 

5.4848 
5.4959 

6.09756 
6.06061 
6.02410 

515.22 
518.36 
521.50 

21124.1 
21382.5 
21642.4 

167 
168 
169 

27889 
28224 
28561 

4657463 
4741632 
4826809 

12.9228 
12.9615 
13.0000 

5.5069 
5.5178 

5.5288 

5.98802 
5.95238 
5.91716 

524.65 
527.79 
530.93 

21904.0 
22167.1 
22431.8 

170 
171 

172 
173 

28900 
29241 
29584 
29929 

4913000 
5000211 
5088448 
5177717 

13.0384 
13.0767 
13.1149 
13.1529 

5.5397 
5.5505 
5.5613 
5.5721 

5.88235 
5.84795 
5.81395 
5.78035 

534.07 
537.21 
540.35 
543.50 

22698.0 
22965.8 
23235.2 
23506.2 

174 
175 
176 

30276 
30625 
30976 

5268024 
5359375 
5451776 

13.1909 
13.2288 
13.2665 

5.5828 
5.5934 
5.6041 

5.74713 
5.71429 

5.68182 

546.64 
549.78 
552.92 

23778.7 
24052.8 
24328.5 

177 
178 
179 

31329 
31684 
32041 

5545233 

5639752 
5735339 

13.3041 
13.3417 
13.3791 

5.6147 
5.6252 
5.6357 

5.64972 
5.61798 
5.58659 

556.06 
559.20 
562.35 

24605.7 
24884.6 
25164.9 

180 

181 
182 
183 

32400 
32761 
33124 
33489 

5832000 
5929741 
6028568 
6128487 

13.4164 
13.4536 
13.4907 
13.5277 

5.6462 
5.6567 
5.6671 
5.6774 

5.55556 
5.52486 
5.49451 
5.46448 

565.49 
568.63 
571.77 
574.91 

25446.9 
25730.4 
26015.5 
26302.2 

184 
185 
186 

33856 
34225 
34596 

6229504 
6331625 
6434856 

13.5647 
13.6015 
13.6382 

5.6877 
5.6980 
5.7083 

5.43478 
5.40541 
5.37634 

578.05 
581.19 
584.34 

26590.4 
26880.3 
27171.6 

187 
188 
189 

34969 
35344 
35721 

6539203 
6644672 
6751269 

13.6748 
13.7113 
13.7477 

5.7185 
5.7287 
5.7388 

5.34759 
5.31915 
5.29101 

587.48 
590.62 
593.76 

27464.6 
27759.1 
28055.2 

190 

191 
192 
193 

36100 
36481 
36864 
37249 

6859000 
6967871 
7077888 
7189057 

13.7840 
13.8203 
13.8564 
13.8924 

5.7489 
5.7590 
5.7690 
5.7790 

5'.  26316 
5.23560 
5.20833 
5.18135 

596.90 
600.04 
603.19 
606.33 

28352.9 
28652.1 
28952.9 
29255.3 

194 
195 
196 

37636 
38025 
38416 

7301384 
7414875 
7529536 

13.9284 
13.9642 
14.0000 

5.7890 
5.7989 

5.8088 

5.15464 
5.12821 
5.10204 

609.47 
612.61 
615  .  75 

29559.2 
29864.8 
30171.9 

197 
198 
199 

38809 
39204 
39601 

7645373 
7762392 
7880599 

14.0357 
14.0712 
14.1067 

5.8186 
5.8285 
5.8383 

5.07614 
5.05051 
5.02513 

618.89 
622  04 
625.18 

30480.5 
30790.7 
31102.6 

56 


FUNCTIONS     OF    THE    NATURAL    NUMBERS 


No. 

Square 

Cube 

Square 
Root 

Cube 
Root 

1000  x 
Recip. 

No.:=Dia. 

Circum. 

Area 

200 

201 
202 
203 

40000 
40401 
40804 
41209 

8000000 
8120601 
8242408 
8365427 

14.1421 
14.1774 
14.2127 
14.2478 

5.8480 
5.8578 
5.8675 
5.8771 

5.00000 
4.97512 
4.95050 
4.92611 

628.32 
631.46 
634.60 
637.74 

31415.9 
31730.9 
32047.4 
32365.5 

204 
205 

206 

41616 
42025 
42436 

8489664 
8615125 
8741816 

14.2829 
14.3178 
14.3527 

5.8868 
5.8964 
5.9059 

4.90196 
4.87805 
4.85437 

640.88 
644  .  03 
647.17 

32685.1 
33006.4 
33329.2 

207 
208 
209 

42849 
43264 
43681 

8869743 
8998912 
9129329 

14.3875 
14.4222 
14.4568 

5.9155 
5.9250 
5.9345 

4.83092 
4.80769 
4.78469 

650.31 
653.45 
656.59 

33653.5 
33979.5 
34307.0 

210 

211 
212 
213 

44100 
44521 
44944 
45369 

9261000 
9393931 
9528128 
9663597 

14.4914 
14.5258 
14.5602 
14.5945 

5.9439 
5.9533 
5.9627 
5.9721 

4.76190 
4.73934 
4.71698 
4.69484 

659.73 
662.88 
666.02 
669.16 

34636.1 
34966.7 
35298.9 
35632.7 

214 
215 
216 

45796 
46225 
46656 

9800344 
9938375 
10077696 

14.6287 
14.6629 
14.6969 

5.9814 
5.9907 
6.0000 

4.67290 
4.65116 
4.62963 

672.30 
675.44 
678.58 

35968.1 
36305.0 
36643.5 

217 
218 
219 

47089 
47524 
47961 

10218313 
10360232 
10503459 

14.7309 
14.7648 
14.7986 

6.0092 
6.0185 
6.0277 

4.60829 
4.58716 
4.56621 

681.73 
684.87 
688.01 

36983.6 
37325.3 
37668.5 

220 
221 
222 
223 

48400 
48841 
49284 
49729 

10648000 
10793861 
10941048 
11089567 

14.8324 
14.8661 
14.899T 
14.9332 

6.0368 
6.0459 
6.0550 
6.0641 

4.54545 
4.52489 
4.50450 
4.48430 

691.15 
694.29 
697.43 
700.58 

38013.3 
38359.6 
38707.6 
39057.1 

224 
225 

226 

50176 
50625 
51076 

11239424 
11390625 
11543176 

14.9666 
15.0000 
15.0333 

6.0732 
6.0822 
6.0912 

4.46429 
4.44444 
4.42478 

703.72 
706.86 
710.00 

39408.1 
39760.8 
40115.0 

227 
228 
229 

51529 
51984 
52441 

11697083 
11852352 
12008989 

15.0665 
15.0997 
15.1327 

6.1002 
6.1091 
6.1180 

4.40529 
4.38596 
4.36681 

713.14 

716.28 
719.42 

40470.8 
40828.1 
41187.1 

230 
231 
232 
233 

52900 
53361 
53824 
54289 

12167000 
12326391 
12487168 
12649337 

15.1658 
15.1987 
15.2315 
15.2643 

6.1269 
6.1358 
6.1446 
6.1534 

4.34783 
4.32900 
4.31034 
4.29185 

722.57 
725.71 
728.85  • 
731.99 

41547.6 
41909.6 
42273.3 
42638.5 

234 
235 
236 

54756 
55225 
55696 

12812904 
12977875 
13144256 

15.2971 
15.3297 
15.3623 

6.1622 
6.1710 
6.1797 

4.27350 
4.25532 
4.23729 

735.13 

738.27 
741.42 

43005.3 
43373.6 
43743.5 

237 
238 
239 

56169 
56644 
57121 

13312053 
13481272 
13651919 

15.3948 
15.4272 
15.4596 

6.1885 
6.1972 
6.2058 

4.21941 
4.20168 
4.18410 

744.56 
747.70 
750  .  84 

44115.0 
44488.1 
44862.7 

240 

241 

243 

57600 
58081 
58564 
59049 

13824000 
13997521 
14172488 
14348907 

15.4919 
15.5242 
15.5563 
15.5885 

6.2145 
6.2231 
6  .  2317 
6.2403 

4.16667 
4.14938 
4.13223 
4.11523 

753.98 
757.12 
760.27 
763.41 

45238.9 
45616.7 
45996.1 
46377.0 

244 
245 
246 

59536 
60025 
60516 

14526784 
14706125 
14886936 

15.6205 
15.6525 
15.6844 

6.2488 
6.2573 
6.2658 

4.09836 
4.08163 
4.06504 

766.55 
769.69 

772.83 

46759.5 
47143.5 
47529.2 

247 
248 
249 

61009 
61504 
62001 

15069223 
15252992 
15438249 

15.7162 
15.7480 
15.7797 

6.2743 
6.2828 
6.2912 

4.04858 
4.03226 
4.01606 

775.97 
779.11 
782.26 

47916.4 
48305.1 
48695.5 

FUNCTIONS     OF    THE    NATURAL    NUMBERS 


57 


No. 

Square 

Cube 

Square 
Root 

Cube 
Root 

1000  x 
Recip. 

Circum. 

Area 

250 
251 
252 
253 

62500 
63001 
63504 
64009 

15625000 
15813251 
16003008 
16194277 

15.8114 
15.8430 
15.8745 
15.9060 

6.2996 
6.3080 
6.3164 
6.3247 

4.00000 
3.98406 
3.96825 
3.95257 

785.40 
788.54 
791.68 

794.82 

49087.4 
49480.9 
49875.9 
50272.6 

254 
255 
256 

64516 
65025 
65536 

16387064 
16581375 
16777216 

15.9374 
15.9687 
16.0000 

6.3330 
6.3413 
6.3496 

3.93701 
3.92157 
3.90625 

797.96 
•801.11 
804.25 

50670.7 
51070.5 
51471.9 

257 
258 
259 

66049 
66564 

67081 

16974593 
17173512 
17373979 

16.0312 
16.0624 
16.0935 

6.3579 
6.3661 
6.3743 

3.89105 
3.87597 
3.86100 

807.39 
810.53 
813.67 

51874.8 
52279.2 
52685.3 

260 

261 
262 
263 

67600 
68121 
68644 
69169 

17576000 
17779581 
17984728 
18191447 

16.1245 
16.1555 
16.1864 
16.2173 

6.3825 
6.3907 
6.3988 
6.4070 

3.84615 
3.83142 
3.81679 

3.80228 

816.81 
819.96 
823.10 
826.24 

53092.9 
53502.1 
53912.9 
54325.2 

264 
265 
266 

69696 
70225 
70756 

18399744 
18609625 
18821096 

16.2481 
16.2788 
16.3095 

6.4151 
6.4232 
6.4312 

3.78788 
3.77358 
3.75940 

829.38 
832.52 
835.66 

54739.1 
55154.6 
55571.6 

267 
268 
269 

71289 
71824 
72361 

19034163 
19248832 
19465109 

16.3401 
16.3707 
16.4012 

6.4393 
6.4473 
6.4553 

3.74532 
3.73134 
3.71747 

838.81 
841.95 
845.09 

55990.2 
56410.4 
56832.2 

270 

271 
272 
273 

72900 
73441 
73984 
74529 

19683000 
19902511 
20123648 
20346417 

16.4317 
16.4621 
16.4924 
16.5227 

6.4633 
6.4713 
6.4792 
6.4872 

3.70370 
3.69004 
3.67647 
3.66300 

848.23 
851.37 
854.51 
857.66 

57255.5 
57680.4 
58106.9 
58534.9 

274 
275 
276 

75076 
75625 
76176 

20570824 
20796875 
21024576 

16.5529 
16.5831 
16.6132 

6.4951 
6.5030 
6.5108 

3.64964 
3.63636 
3.62319 

860.80 
863.94 
867.08 

58964.6 
59395.7 
59828.5 

277 
278 
279 

76729 

77284 
77841 

21253933 
21484952 
21717639 

16.6433 
16.6733 
16.7033 

6.5187 
6.5265 
6.5343 

3.61011 
3.59712 
3.58423 

870.22 
873.36 
876.50 

60262.8 
60698.7 
61136.2 

280 

281 
282 
283 

78400 
78961 
79524 
80089 

21952000 
22188041 
22425768 
22665187 

16.7332 
16.7631 
16.7929 
16.8226 

6.5421 
6.5499 
6.5577 
6.5654 

3.57143 
3.55872 
3.54610 
3.53357 

879.65 
882.79 
885.93 
889.07 

61575.2 
62015.8 
62458.0 
62901.8 

284 
285 
286 

80656 
81225 
81796 

22906304 
23149125 
23393656 

16.8523 
16.8819 
16.9115 

6.5731 

6.5808 
6.5885 

3.52113 
3.50877 
3.49650 

892.21 
895.35 
898.50 

63347.1 
63794.0 
64242.4 

287 
288 
289 

82369 
82944 
83521 

23639903 

23887872 
24137569 

16.9411 
16.9706 
17.0000 

6.5962 
6.6039 
6.6115 

3.48432 
3.47222 
3.46021 

901.64 
904.78 
907.92 

64692.5 
65144.1 
65597.2 

290 

291 
292 
293 

84100 
84681 
85264 
85849 

24389000 
24642171 
24897088 
25153757 

17.0294 
17.0587 
17.0880 
17.1172 

6.6191 
6.6267 
6.6343 
6.6419 

3.44828 
3.43643 
3.42466 
3.41297 

911.06 
914.20 
917.35 
920.49 

66052.0 
66508.3 
66966.2 
67425.6 

294 
295 
296 

86436 
87025 
87616 

25412184 
25672375 
25934336 

17.1464 
17.1756 
17.2047 

6.6494 
6.6569 
6.6644 

3.40136 
3.38983 
3.37838 

923.63 

926.77 
929.91 

67886.7 
68349.3 
68813.4 

297 
298 
299 

88209 
88804 
89401 

26198073 
26463592 
26730899 

17.2337 

17.2627 
17.2916 

6.6719 
6.6794 
6.6869 

3.36700 
3.35570 
3.34448 

933.05 
936.19 
939.34 

69279.2 
69746.5 
70215.4 

FUNCTIONS     OF    THE    NATURAL    NUMBERS 


No. 

Sauare 

Cube 

Square 
Root 

Cube 
Root 

1000  x 
Recip. 

No.=Dia. 

Circurn. 

Area 

300 

301 
302 
303 

90000 
90601 
91204 
91809 

27000000 
27270901 
27543608 
27818127 

17.3205 
17.3494 
17.3781 
17.4069 

6.6943 
6.7018 
6.7092 
6.7166 

3.33333 
3.32226 
3.31126 
3.30033 

942.48 
945.62 
948.76 
951.90 

70685.8 
71157.9 
71631.5 
72106.6 

304 
305 

306 

92416 
93025 
93636 

28094464 
28372625 
28652616 

17.4356 
17.4642 
17.4929 

6.7240 
6.7313 
6.7387 

3.28947 
3.27869 
3.26797 

955.04 
958.19 
961.33 

72583.4 
73061.7 
73541.5 

307 
308 
309 

94249 
94864 
95481 

28934443 
29218112 
29503629 

17.5214 
17.5499 
17.5784 

6.7460 
6.7533 
6.7606 

3.25733 
3.24675 
3.23625 

964.47 
967.61 
970.75 

74023.0 
74506.0 
74990.6 

310 

311 
312 
313 

96100 
96721 
97344 
97969 

29791000 
30080231 
30371328 
30664297 

17.6068 
17.6352 
17.6635 
17.6918 

6.7679 
6.7752 
6.7824 
6.7897 

3.22581 
3.21543 
3.20513 
3.19489 

973.89 
977.04 

980.18 
983.32 

75476.8 
75964.5 
76453.8 
76944.7 

314 
315 
316 

98596 
99225 
99856 

30959144 
31255875 
31554496 

17.7200 
17.7482 
17.7764 

6.7969 
6.8041 
6.8113 

3.18471 
3.17460 
3.16456 

986.46 
989.60 
992.74 

77437.1 
77931.1 
78426.7 

317 
318 
319 

100489 
101124 
101761 

31855013 
32157432 
32461759 

17.8045 
17.8326 
17.8606 

6.8185 
6.8256 
6.8328 

3.15457 
3.14465 
3.13480 

995.88 
999.03 
1002.2 

78923.9 
79422.6 
79922.9 

320 
321 
322 
323 

102400 
103041 
103684 
104329 

32768000 
33076161 
33386248 
33698267 

17.8885 
17.9165 
17.9444 
17.9722 

6.8399 
6.8470 
6.8541 
6.8612 

3.12500 
3.11527 
3.10559 
3.09598 

1005.3 
1008.5 
1011.6 
1014.7 

80424.8 
80928.2 
81433.2 
81939.8 

324 
325 

326 

104976 
105625 
106276 

34012224 
34328125 
34645976 

18.0000 
18.0278 
18.0555 

6.8683 
6.8753 
6.8824 

3.08642 
3.07692 
3.06748 

1017.9 
1021.0 
1024.2 

82448.0 

82957.7 
83469.0 

327 
328 
329 

106929 
107584 
108241 

34965783 
35287552 
35611289 

18.0831 
18.1108 
18.1384 

6.8894 
6.8964 
6.9034 

3.05810 
3.04878 
3.03951 

1027.3 
1030.4 
1033.6 

83981.8 
84496.3 
85012.3 

330 
331 
332 
333 

108900 
109561 
110224 
110889 

35937000 
36264691 
36594368 
36926037 

18.1659 
18.1934 
18.2209 
18.2483 

6.9104 
6.9174 
6.9244 
6.9313 

3.03030 
3.02115 
3.01205 
3.00300 

1036.7 
1039.9 
1043.0 
1046.2 

85529.9 
86049.0 
86569.7 
87092.0 

334 
335 
336 

111556 
112225 
112896 

37259704 
37595375 
37933056 

18.2757 
18.3030 
18.3303 

6.9382 
6.9451 
6.9521 

2.99401 
2.98507 
2.97619 

1049.3 
1052.4 
1055.6 

87615.9 
88141.3 
88668.3 

337 
338 
339 

113569 
114244 
114921 

38272753 
38614472 
38958219 

18.3576 
18.3848 
18.4120 

6.9589 
6.9658 
6.9727 

2.96736 

2.95858 
2.94985 

1058.7 
1061.9 
1065.0 

89196.9 
89727.0 
90258.7 

340 
341 
342 
343 

115600 
116281 
116964 
117649 

39304000 
39651821 
40001688 
40353607 

18.4391 
18.4662 
18.4932 
18.5203 

6.9795 
6.9864 
6.9932 
7.0000 

2.94118 
2.93255 
2.92398 
2.91545 

1068.1 
1071.3 
1074.4 
1077.6 

90792.0 
91326.9 
91863.3 
92401.3 

344 

345 
346 

118336 
119025 
119716 

40707584 
41063625 
41421736 

18.5472 
18.5742 
18.6011 

7.0068 
7.0136 
7.0203 

2.90698 
2.89855 
2.89017 

1080.7 
1083.8 
1087.0 

92940.9 
93482.0 
94024.7 

347 
348 
349 

120409 
121104 
121801 

41781923 
42144192 
42508549 

18.6279 
18.6548 
18.6815 

7.0271 
7.0338 
7.0406 

2.88184 
2.87356 
2.86533 

1090.1 
1093.3 
1096.4 

94569.0 
95114.9 
95662.3 

FUNCTIONS     OF    THE    NATURAL    NUMBERS 


59 


No. 

Square 

Cube 

Square 
Root 

Cube 
Root 

1000  x 
Recip. 

Circum. 

Area 

350 
351 
352 
353 

122500 
123201 
123904 
124609 

42875000 
43243551 
43614208 
43986977 

18.7083 
18.7350 
18.7617 
18.7883 

7.0473 
7.0540 
7.0607 
7.0674 

2.85714 

2.84900 
2.84091 
2.83286 

1099.6 
1102.7. 
1105.8 
1109.0 

96211.3 
96761.8 
97314.0 

97867.7 

354 
355 
356 

125316 
126025 
126736 

44361864 
44738875 
45118016 

18.8149 
18.8414 
18.8680 

7.0740 
7.0807 
7.0873 

2.82486 
2.81690 
2.80899 

1112.1 
1115.3 
1118.4 

98423.0 
98979.8 
99538.2 

357 

358 
359 

127449 
128164 

128881 

45499293 
45882712 
46268279 

18.8944 
18.9209 
18.9473 

7.0940 
7.1006 
7.1072 

2.80112 
2.79330 

2.78552 

1121.5 
1124.7 
1127.8 

100098 
100660 
101223 

360 
361 
362 
363 

129600 
130321 
131044 
131769 

46656000 
47045881 
47437928 
47832147 

18.9737 
19.0000 
19.0263 
19.0526 

7.1138 
7.1204 
7.1269 
7.1335 

2.77778 
2.77008 
2.76243 
2.75482 

1131.0 
1134.1 
1137.3 
1140.4 

101788 
102354 
102922 
103491 

364 
365 
366 

132496 
133225 
133956 

48228544 
48627125 
49027896 

19.0788 
19.1050 
19.1311 

7.1400 
7.1466 
7.1531 

2.74725 
2.73973 
2.73224 

1143.5 
1146.7 
1149.8 

104062 
104635 
105209 

367 
368 
369 

134689 
135424 
136161 

49430863 
49836032 
50243409 

19.1572 
19.1833 
19.2094 

7.1596 
7.1661 
7.1726 

2.72480 
2.71739 
2.71003 

1153.0 
1156.1 
1159.2 

105785 
106362 
106941 

370 
371 
372 
373 

136900 
137641 
138384 
139129 

50653000 
51064811 
51478848 
51895117 

19.2354 
19.2614 
19.2873 
19.3132 

7.1791 
7.1855 
7.1920 
7.1984 

2.70270 
2.69542 
2.68817 
2.68097 

1162.4 
1165.5 
1168.7 
1171.8 

107521 
108103 
108687 
109272 

374 
375 
376 

139876 
140625 
141376 

52313624 
52734375 
53157376 

19.3391 
19.3649 
19.3907 

7.2048 
7.2112 
7.2177 

2.67380 
2.66667 
2.65957 

1175.0 

1178.1 
1181.2 

109858 
110447 
111036 

377 
378 
379 

142129 
142884 
143641 

53582633 
54010152 
54439939 

19.4165 
19.4422 
19.4679 

7.2240 
7.2304 
7.2368 

2.65252 
2.64550 
2.63852 

1184.4 
1187.5 
1190.7 

111628 
112221 
112815 

380 
381 
382 
383 

144400 
145161 
145924 
146689 

54872000 
55306341 
55742968 
56181887 

19.4936 
19.5192 
19.5448 
19.5704 

7.2432 

7.2495 
7.2558 
7.2622 

2.63158 
2.62467 
2.61780 
2.61097 

1193.8 
1196.9 
1200.1 
1203.2 

113411 
114009 
114608 
115209 

384 
385 
386 

147456 
148225 
148996 

56623104 
57066625 
57512456 

19.5959 
19.6214 
19.6469 

7.2685 
7.2748 
7.2811 

2.60417 
2.59740 
2.59067 

1206.4 
1209.5 
1212.7 

115812 
116416 
117021 

387 
388 
389 

149769 
150544 
151321 

57960603 
58411072 
58863869 

19.6723 
19.6977 
19.7231 

7.2874 
7.2936 
7.2999 

2.58398 
2.57732 
2.57069 

1215.8 
1218.9 
1221.1 

117628 
118237 
118847 

390 
391 
392 
393 

152100 
152881 
153664 
154449 

59319000 
59776471 
60236288 
60698457 

19.7484 
19.7737 
19-7990 
19.8242 

7.3061 
7.3124 
7.3186 
7.3248 

2.56410 
2.55755 
2.55102 
2.54453 

1225.2 
1228.4 
1232.5 
1234.6 

119459 
120072 
120687 
121304 

394 
395 
396 

155236 
156025 
156816 

61162984 
61629875 
62099136 

19.8494 
19.8746 
19.8997 

7.3310 
7.3372 
7.3434 

2.53807 
2.53165 
2.52525 

1237.8 
1240.9 
1244.1 

121922 
122542 
123163 

397 
398 
399 

157609 
158404 
159201 

62570773 
63044792 
63521199 

19.9249 
19.9499 
19.9750 

7.3496 
7.3558 
7.3619 

2.51889 
2.51256 
2.50627 

1247.2 
1250.4 
1253.5 

123786 
124410 
125036 

60 


FUNCTIONS     OF    THE    NATURAL    NUMBERS 


No. 

Square 

Cube 

Square 
Root 

Cube 
Root 

1000  x 
Recip. 

No.=: 

£Dia. 

Circum. 

Area 

400 
401 
402 
403 

160000 
160801 
161604 
162409 

64000000 
64481201 
64964808 
65450827 

20.0000 
20.0250 
20.0499 
20.0749 

7.3681 
7.3742 
7.3803 
7.3864 

2.50000 
2.49377 
2.48756 
2.48139 

1256.6 
1259.8 
1262.9 
1266.1 

125664 
126293 
126923 
127556 

404 
405 
406 

163216 
164025 
164836 

65939264 
66430125 
66923416 

20.0998 
20.1246 
20.1494 

7.3925 
7.3986 

7.4047 

2.47525 
2.46914 
2.46305 

1269.2 
1272.3 
1275.5 

128190 
128825 
129462 

407 
408 
409 

165649 
166464 
167281 

67419143 
67917312 
68417929 

20.1742 
20.1990 
20.2237 

7.4108 
7.4169 
7.4229 

2.45700 
2.45098 
2.44499 

1278.6 
1281.8 
1284.9 

130100 
130741 
131382 

410 
411 
412 
413 

168100 
168921 
169744 
170569 

68921000 
69426531 
69934528 
70444997 

20.2485 
20.2731 
20.2978 
20.3224 

7.4290 
7.4350 
7.4410 
7.4470 

2.43902 
2.43309 
2.4271S 
2.42131 

1288.1 
1291.2 
1294.3 
1297.5 

132025 
132670 
133317 
133965 

414 
415 
416 

171396 
172225 
173056 

70957944 
71473375 
71991296 

20.3470 
20.3715 
20.3961 

7.4530 
7.4590 
7.4650 

2.41546 
2.40964 
2.40385 

1300.6 
1303.8 
1306.9 

134614 
135265 
135918 

417 
418 
419 

173889 
174724 
175561 

72511713 
73034632 
73560059 

20.4206 
20.4450 
20.4695 

7.4710 
7.4770 
7.4829 

2.39808 
2.39234 
2.38663 

1310.0 
1313.2 
1316.3 

136572 
137228 
137885 

420 
421 
422 
423 

176400 
177241 
178084 
178929 

74088000 
74618461 
75151448 
75686967 

20.4939 
20.5183 
20.5426 
20.5670 

7.4889 
7.4948 
7.5007 
7.5067 

2.38095 
2.37530 
2.36967 
2.36407 

1319.5 
1322.6 
1325.8 
1328.9 

138544 
139205 
139867 
140531 

424 
425 
426 

179776 
180625 
181476 

76225024 
76765625 
77308776 

20.5913 
20.6155 
20.6398 

7.5126 
7.5185 
7.5244 

2.35849 
2.35294 
2.34742 

1332.0 
1335.2 
1338.3 

141196 
141863 
142531 

427 
428 
429 

182329 
183184 
184041 

77854483 

78402752 
78953589 

20.6640 
20.6882 
20.7123 

7.5302 
7.5361 
7.5420 

2.34192 
2.33645 
2.33100 

1341.5 
1344.6 
1347.7 

143201 
143872 
144545 

430 
431 
432 
433 

184900 
185761 
186624 
187489 

79507000 
80062991 
80621568 
81182737 

20.7364 
20.7605 

20.7846 
20.8087 

7.5478 
7.5537 
7.5595 
7.5654 

2.32558 
2.32019 
2.31481 
2.30947 

1350.9 
1354.0 
1357.2 
1360.3 

145220 
145896 
146574 
147254 

434 
435 
436 

188356 
189225 
190096 

81746504 

82312875 
82881856 

20.8327 
20.8567 
20.8806 

7.5712 
7.5770 

7.5828 

2.30415 

2.29885 
2.29358 

1363.5 
1366.6 
1369.7 

147934 
148617 
149301 

437 
438 
439 

190969 
191844 
192721 

83453453 

84027672 
84604519 

20.9045 
20.9284 
20.9523 

7.5886 
7.5944 
7.6001 

2.28833 
2.28311 
2.27790 

1372.9 
1376.0 
1379.2 

149987 
150674 
151363 

440 
441 
442 
443 

193600 
194481 
195364 
196249 

85184000 
85766121 
86350888 
86938307 

20.9762 
21.0000 
21.0238 
21.0476 

7.6059 
7.6117 
7.6174 
7.6232 

2.27273 

2.26757 
2.26244 
2.25734 

1382.3 
1385.4 
1388.6 
1391.7 

152053 
152745 
153439 
154134 

444 
445 
446 

197136 
198025 
198916 

87528384 
88121125 
88716536 

21.0713 
21.0950 
21.1187 

7.6289 
7.6346 
7.6403 

2.25225 
2.24719 
2.24215 

1394.9 
1398.0 
1401.2 

154830 
155528 
156228 

447 
448 
449 

199809 
200704 
201601 

89314623 
89915392 
90518849 

21.1424 
21.1660 
21.1896 

7.6460 
7.6517 
7.6574 

2.23714 
2.23214 

2.22717 

1404.3 
1407.4 
1410.6 

156930 
157633 
158337 

FUNCTIONS    OF    THE    NATURAL    NUMBERS 


No. 

Square 

Cube 

Square 
Root 

Cube 
Root 

1000  x 
Recip. 

No- 

:Dia. 

Circum. 

Area 

450 
451 
452 
453 

202500 
203401 
204304 
205209 

91125000 
91733851 
92345408 
92959677 

21.2132 
21.2368 
21.2603 
21.2838 

7.6631 

7.6688 
7.6744 
7.6801 

2.22222 
2.21729 
2.21239 
2.20751 

1413.7 
1416.9 
1420.0 
1423.1 

159043 
159751 
160460 
161171 

454 
455 
456 

206116 
207025 
207936 

93576664 
94196375 
94818816 

21.3073 
21.3307 
21.3542 

7.6857 
7.6914 
7.6970 

2.20264 
2.19780 
2.19298 

1426.3 
1429.4 
1432.6 

161883 
162597 
163313 

457 
458 
459 

208849 
209764 
210681 

95443993 
96071912 
96702579 

21.3776 

21.4009 
21.4243 

7.7026 
7.7082 
7.7138 

2.18818 
2.18341 
2.17865 

1435.7 
1438.9 
1442.0 

164030 
164748 
165468 

460 
461 
462 
463 

211600 
212521 
213444 
214369 

97336000 
97972181 
98611128 
99252847 

21.4476 
21.4709 
21.4942 
21.5174 

7.7194 
7.7250 
7.7306 
7.7362 

2.17391 
2.16920 
2.16450 
2.15983 

1445.1 

1448.3 
1451.4 
1454.6 

166190 
166914 
167639 
168365 

464 
465 
466 

215296 
216225 
217156 

99897344 
100544625 
101194696 

21.5407 
21.5639 
21.5870 

7.7418 
7.7473 
7.7529 

2.15517 
2.15054 
2.14592 

1457.7 

1460.8 
1464.0 

169093 
169823 
170554 

467 
468 
469 

218089 
219024 
219961 

101847563 
102503232 
103161709 

21.6102 
21.6333 
21.6564 

7.7584 
7.7639 
7.7695 

2.14133 
2.13675 
2.13220 

1467.1 
1470.3 
1473.4 

171287 
172021 
172757 

470 
471 

472 
473 

220900 
221841 
222784 
223729 

103823000 
104487111 
105154048 
105823817 

21.6795 
21.7025 
21.7256 
21.7486 

7.7750 

7.7805 
7.7860 
7.7915 

2.12766 
2.12314 
2.11864 
2.11416 

1476.5 
1479.7 
1482.8 
1486.0 

173494 
174234 
174974 
175716 

474 

475 
476 

224676 
225625 
226576 

106496424 
107171875 
107850176 

21.7715 
21.7945 
21.8174 

7.7970 
7.8025 
7.8079 

2.10970 
2.10526 
2.10084 

1489.1 
1492.3 
1495.4 

176460 
177205 
177952 

477 
478 
479 

227529 
228484 
229441 

108531333 
109215352 
109902239 

21.8403 
21.8632 
21.8861 

7.8134 
7.8188 
7.8243 

2.09644 
2.09205 
2.08768 

1498.5 
1501.7 
1504,8 

178701 
179451 
180203 

480 

481 
482 
483 

230400 
231361 
232324 
233289 

110592000 
111284641 
111980168 
112678587 

21.9089 
21.9317 
21.9545 
21.9773 

7.8297 
7.8352 
7.8406 
7.8460 

2.08333 
2.07900 
2.07469 
2.07039 

1508.0 
1511.1 
1514.2 
1517.4 

180956 
181711 
182467 
183225 

484 
485 

486 

234256 
235225 
236196 

113379904 
114084125 
114791256 

2 

.0000 
.0227 
.0454 

7.8514 

7.8568 
7.8622 

2.06612 
2.00186 
2.05761 

1520.5 
1523.7 
1526.8 

183984 
184745 
185508 

487 
488 
489 

237169 
238144 
239121 

115501303 
116214272 
116930169 

2 

.0681 
.0907 
.1133 

7.8676 
7.8730 
7.8784 

2.05339 
2.04918 
2.04499 

1530.0 
1533.1 
1536.2 

186272 
187038. 

187805. 

490 
491 
492 
493 

240100 
241081 
242064 
243049 

117649000 
118370771 
119095488 
119823157 

2 
2 

.1359 
.1585 
.1811 
.2036 

7.8837 
7.8891 
7.8944 
7.8998 

2.04082 
2.03666 
2.03252 

2.02840 

1539.4 
1542.5 
1545.7 
1548.8 

188574 
189345 
190117 
190890 

494 
495 
496 

244036 
245025 
246016 

120553784 
121287375 
-  122023936 

2 

.2261 
.2486 
.2711 

7.9051 
7.9105 
7.9158 

2.02429 
2.02020 
2.01613 

1551.9 
1555.1 
1558.2 

191665 
192442 
193221 

497 
498 
499 

247009 
248004 
249001 

122763473 
123505992 
124251499 

2 

.2935 
.3159 
.3383 

7.9211 
7.9264 
7.9317 

2.01207 
2.00803 
2.00401 

1561.4 
1564.5 
1567.7 

194000 
194782 
195565 

FUNCTIONS  OF  THE  NATURAL  NUMBERS 


No. 

Square 

Cube 

Square 
Root 

Cube 
Root 

1000  x 
Recip. 

No.—  Dia. 

Circum. 

Area 

500 

501 
502 
503 

250000 
251001 
252004 
253CT09 

125000000 
125751501 
126506008 
127263527 

22.3607 
22.3830 
22.4054 
22.4277 

7.9370 
7.9423 
7.9476 
7.9528 

2.00000 
1.99601 
1.99203 
1.98807 

1570.8 
1573.9 
1577.1 
1580.2 

196350 
197136 
197923 
198713 

504 
505 
506 

254016 
255025 
256036 

128024064 
128787625 
129554216 

22.4499 
22  4722 
22.4944 

7.9581 
7.9634 
7.9686 

1.98413 
1.98020 
1.97628 

1583.4 
1586.5 
1589.6 

199504 
200296 
201090 

507 
508 
509 

257049 
258064 
259081 

130323843 
131096512 
131872229 

22.5167 
22.5389 
22.5610 

7.9739 
7.9791 
7.9843 

1.97239 
1.96850 
1.96464 

1592.8 
1595.9 
1599.1 

201886 
202683 
203482 

510 

511 
512 
513 

260100 
261121 
262144 
263169 

132651000 
133432831 
134217728 
135005697 

22.5832 
22.6053 
22.6274 
22.6495 

7.9896 
7.9948 
8.0000 
8.0052 

1.96078 
1.95695 
1.95312 
1.94932 

1602.2 
1605.4 
1608.5 
1611.6 

204282 
205084 
205887 
206692 

514 
515 
516 

264196 
265225 
266256 

135796744 
136590875 
137388096 

22.6716 
22.6936 
22.7156 

8.0104 
8.0156 
8.0208 

1.94553 
1.94175 
1.93798 

1614.8 
1617.9 
1621.1 

207499 
208307 
209117 

517 
518 
519 

267289 
268324 
269361 

138188413 
138991832 
139798359 

22.7376 
22.7596 
22.7816 

8.0260 
8.0311 
8.0363 

1.93424 
1.93050 
1.92678 

1624.2 
1627.3 
1630.5 

20992S 
210741 
211556 

520 

521 
522 
523 

270400 
271441 
272484 
273529 

140608000 
141420761 
142236648 
143055667 

22.8035 
22.8254 
22.8473 
22.8692 

8.0415 
8.0466 
8.0517 
8.0569 

1.92308 
1.91939 
1.91571 
1.91205 

1633.6 
1636.8 
1639.9 
1643.1 

212372 
213189 
214008 
214829 

524 
525 
526 

274576 
275625 
276676 

143877824 
144703125 
145531576 

22.8910 
22.9129 
22.9347 

8.0620 
8.0671 
8.0723 

1.90840 
1.90476 
1.90114 

1646.2 
1649.3 
1652.5 

215651 
216475 
217301 

527 
528 
529 

277729 

278784 
279841 

146363183 
147197952 
148035889 

22.9565 
22.9783 
23.0000 

8.0774 
8.0825 
8.0876 

1.89753 
1.89394 
1.89036 

1655.6 
1658.8 
1661.9 

218128 
218956 
219787 

530 
531 
532 
533 

280900 
281961 
283024 
284089 

148877000 
149721291 
150568768 
151419437 

23.0217 
23.0434 
23.0651 
23.0868 

8.0927 

8.0978 
8.1028 
8.1079 

1.88679 
1.88324 
1.87970 
1.87617 

1665.0 
1668.2 
1671.3 
1674.5 

220618 
221452 
222287 
223123 

534 
535 

536 

285156 
286225 
287296 

152273304 
153130375 
153990656 

23.1084 
23.1301 
23.1517 

8.1130 
8.1180 
8.1231 

1.87266 
1.86916 
1.86567 

1677.6 
1680.8 
1683.9 

223961 
224801 
225642 

537 

538 
539 

288369 
289444 
290521 

154854153 
155720872 
156590819 

23.1733 
23.1948 
23.2164 

8.1281 
8.1332 
8.1382 

1.86220 
1.85874 
1.85529 

1687.0 
1690.2 
1693.3 

226484 
227329 
228175 

540 
541 
542 
543 

291600 
292681 
293764 
294849 

157464000 
158340421 
159220088 
160103007 

23.2379 
23.2594 
23.2809 
23.3024 

8.1433 

8.1483 
8.1533 
8.1583 

1.85185 
1.84843 
1.84502 
1.84162 

1696.5 
1699.6 
1702.7 
1705.9 

229022 
229871 
230722 
231574 

544 
545 
546 

295936 
297025 
298116 

160989184 
161878625 
162771336 

23.3238 
23.3452 
23.3666 

8.1633 
8.1683 
8.1733 

1.83824 
1.83486 
1.83150 

1709.0 
1712.2 
1715.3 

232428 
233283 
234140 

547 

548 
549 

299209 
300304 
301401 

163667323 
164566592 
165469149 

23.3880 
23.4094 
23.4307 

8.1783 
8.1833 
8.1882 

1.82815 
1.82482 
1.82149 

1718.5 
1721.6 
1724.7 

234998 
235858 
236720 

FUNCTIONS    OF    THE    NATURAL    NUMBERS 


68 


No. 

Square 

Cube 

Square 
Root 

Cube 
Root 

1000  x 
Recip. 

No.^Dia. 

Circum. 

Area 

550 

302500 

166375000 

23.4521 

8.1932 

1.81818 

1727.9 

237583 

551 

303601 

167284151 

23.4734 

8.1982 

1.81488 

1731.0 

238448 

552 

304704 

168196608 

23.4947 

8.2031 

1.81159 

1734.2 

239314 

553 

305809 

169112377 

23.5160 

8.2081 

1.80832 

1737.3 

240182 

554 

306916 

170031464 

23.5372 

8.2130 

1.80505 

1740.4 

241051 

555 

308025 

170953875 

23.5584 

8.2180 

1.80180 

1743.6 

241922 

556 

309136 

171879616 

23.5797 

8.2229 

1.79856 

1746.7 

242795 

557 

310249 

172808693 

23.6008 

8.2278 

1.79533 

1749.9 

243669 

558 

311364 

173741112 

23.6220 

8.2327 

1.79211 

1753.0 

244545 

559 

312481 

174676879 

23.6432 

8.2377 

1.78891 

1756.2 

245422 

560 

313600 

175616000 

23.6643 

8.2426 

1.78571 

1759.3 

246301 

561 

314721 

176558481 

23.6854 

8.2475 

1.78253 

1762.4 

247181 

562 

315844 

177504328 

23.7065 

8.2524 

1.77936 

1765.6 

248063 

563 

316969 

178453547 

23.7276 

8.2573 

1.77620 

1768.7 

248947 

564 

318096 

179406144 

23.7487 

8.2621 

1.77305 

1771.9 

249832 

565 

319225 

180362125 

23.7697 

8.2670 

1.76991 

1775.0 

250719 

566 

320356 

181321496 

23.7908 

8.2719 

1.76678 

1778.1 

251607 

567 

321489 

182284263 

23.8118 

8.2768 

1.76367 

1781.3 

252497 

568 

322624 

183250432 

23.8328 

8.2816 

1.76056 

1784.4 

253388 

569 

323761 

184220009 

23.8537 

8.2865 

1.75747 

1787.6 

254281 

570 

324900 

185193000 

23.8747 

8.2913 

1.75439 

1790.7 

255176 

571 

326041 

186169411 

23.8956 

8.2962 

1.75131 

1793.8 

256072 

572 

327184 

187149248 

23.9165 

8.3010 

1.74825 

1797.0 

256970 

573 

328329 

188132517 

23.9374 

8.3059 

1.74520 

1800.1 

257869 

574 

329476 

189119224 

23.9583 

8.3107 

1.74216 

1803.3 

258770 

575 

330625 

190109375 

23.9792 

8.3155 

1.73913 

1806.4 

259672 

576 

331776 

191102976 

24.0000 

8.3203 

1.73611 

1809.6 

260576 

577 

332929 

192100033 

24.0208 

8.3251 

1.73310 

1812.7 

261482 

578 

334084 

193100552 

24.0416 

8.3300 

1.73010 

1815.8 

262389 

579 

335241 

194104539 

24.0624 

8.3348 

1.72712 

1819.0 

263298 

580 

336400 

195112000 

24.0832 

8.3396 

1.72414 

1822.1 

264208 

581 

337561 

196122941 

24.1039 

8.3443 

1.72117 

1825.3 

265120 

582 

338724 

197137368 

24.1247 

8.3491 

1.71821 

1828.4 

266033 

583 

339889 

198155287 

24.1454 

8.3539 

1.71527 

1831.6 

266948 

584 

341056 

199176704 

24.1661 

8.3587 

1.71233 

1834.7 

267865 

585 

342225 

200201625 

24.1868 

8.3634 

1.70940 

1837.8 

268783 

586 

343396 

201230056 

24.2074 

8.3682 

1.70648 

1841.0 

269703 

587 

344569 

202262003 

24.2281 

8.3730 

1.70358 

1844.1 

270624 

588 

345744 

203297472 

24.2487 

8.3777 

1.70068 

1847.3 

271547 

589 

346921 

204336469 

24.2693 

8.3825 

1.69779 

1850.4 

272471 

590 

348100 

205379000 

24.2899 

8.3872 

1.69492 

1853.5 

273397 

591 

349281 

206425071 

24.3105 

8.3919 

1.69205 

1856.7 

274325 

592 

350464 

207474688 

24.3311 

8.3967 

1.68919 

1859.8 

275254 

593 

351649 

208527857 

24.3516 

8.4014 

1.68634 

1863.0 

276184 

594 

352836 

209584584 

24.3721 

8.4061 

1.68350 

1866.1 

277117 

595 

354025 

210644875 

24.3926 

8.4108 

1.68067 

1869.2 

278051 

596 

355216 

211708736 

24.4131 

8.4155 

1.67785 

1872.4 

278986 

597 

356409 

212776173 

24.4336 

8.4202 

1.67504 

1875.5 

279923 

598 

357604 

213847192 

24.4540 

8.4249 

1.67224 

1878.7 

280S62 

599 

358801 

214921799 

24.4745 

8.4296 

1.66945 

1881.8 

281802 

64 


FUNCTIONS  OF  THE  NATURAL  NUMBERS 


No. 

Square 

Cube 

Square 
Root 

Cube 
Root 

1000  x 
Recip. 

No.=Dia. 

Circum. 

Area 

600 

601 
602 
603 

360000 
361201 
362404 
363609 

216000000 
217081801 
218167208 
219256227 

24.4949 
24.5153 
24.5357 
24.5561 

8.4343 
8.4390 
8.4437 
8.4484 

I 

1 
1 
1 

.66667 
.66389 
.66113 
.65837 

1885.0 
1888.1 
1891.2 
1894.4 

282743 

2836S7 
284631 
285578 

604 
605 

606 

364816 
366025 
367236 

220348864 
221445125 
222545016 

24.5764 
24.5967 
24.6171 

8.4530 
8.4577 
8.4623 

1 
1 

1 

.65563 
.65289 
.65017 

1897.5 
1900.7 
1903.8 

286526 

287475 
288426 

607  ' 
608 
609 

368449 
369664 

370881 

223648543 
224755712 
225866529 

24.6374 

24.6577 
24.6779 

8.4670 
8.4716 
8.4763 

1.64745 
1.64474 
1.64204 

1906.9 
1910.1 
1913.2 

289379 
290333 
291289 

610 

611 
612 
613 

372100 
373321 
374544 
375769 

226981000 
228099131 
229220928 
230346397 

24.6982 
24.7184 
24.7386 
24.7588 

8.4809 
8.4856 
8.4902 
8.4948 

1 
1 
1 
1 

.63934 
.63666 
.63399 
.63132 

1916.4 
1919.5 

1922.7 
1925.8 

292247 
293206 
294166 
295128 

614 
615 
616 

376996 
378225 
379456 

231475544 
232608375 
233744896 

24.7790 
24.7992 
24.8193 

8.4994 
8.5040 
8.5086 

1 

1 

1 

.62866 
.62602 
.62338 

1928.9 
1932.1 
1935.2 

296092 
297057 
298024 

617 
618 
619 

380689 
381924 
383161 

234885113 
236029032 
237176659 

24.8395 
24.8596 
24.8797 

8.5132 
8.5178 
8.5224 

1 

1 
1 

.62075 
.61812 
.61551 

1938.4 
1941.5 
1944.6 

298992 
299962 
300934 

620 
621 
622 
623 

384400 
385641 
386884 
388129 

238328000 
239483061 
240641848 
241804367 

24.8998 
24.9199 
24.9399 
24.9600 

8.5270 
8.5316 
8.5362 
8.5408 

1 
1 
1 
1 

.61290 
.61031 
.60772 
.60514 

1947.8 
1950.9 
1954.1 
1957.2 

301907 
302882 
303858 
304836 

624 
625 
626 

389376 
390625 
391876 

242970624 
244140625 
245314376 

24.9800 
25.0000 
25.0200 

8.5453 
8.5499 
8.5544 

1 
1 

1 

.60256 
.60000 
.59744 

1960.4 
1963.5 
1966.6 

305815 
306796 
307779 

627 
628 
629 

393129 
394384 
395641 

246491883 
247673152 

248858189 

25.0400 
25.0599 
25.0799 

8.5590 
8.5635 
8.5681 

1 
1 

1 

.59490 
.59236 
.58983 

1969.8 
1972.9 
1976.1 

308763 
309748 
310736 

630 
631 
632 
633 

396900 
398161 
399424 
400689 

250047000 
251239591 
252435968 
253636137 

25.0998 
25.1197 
25.1396 
25.1595 

8.5726 
8.5772 
8.5817 
8.5862 

1 

1 
1 
1 

.58730 
.58479 
.58228 
.57978 

1979.2 
1982.4 
1985.5 
1988.6 

311725 
312715 
313707 
314700 

634 
635 
636 

401956 
403225 
404496 

254840104 
256047875 
257259456 

25.1794 
25.1992 
25.2190 

8.5907 
8.5952 
8.5997 

1 
I 

1 

.57729 
.57480 
.57233 

1991.8 
1994.9 
1998.1 

315696 
316692 
317690 

637 
638 
639 

405769 
407044 
408321 

258474853 
259694072 
260917119 

25.2389 
25.2587 
25.2784 

8.6043 
8.6088 
8.6132 

1 
1 
1 

.56986 
.56740 
.56495 

2001.2 
2004.3 
2007.5 

318690 
319692 
320695 

640 
641 
642 
643 

409600 
410881 
412164 
413449 

262144000 
263374721 
264609288 
265847707 

25.2982 
25.3180 
25.3377 
25.3574 

8.6177 
8.6222 
8.6267 
8.6312 

1 
1 
1 
1 

.56250 
.56006 
.55763 
.55521 

2010.6 
2013.8 
2016.9 
2020.0 

321699" 
322705 
323713 
324722 

644 
645 
646 

414736 
416025 
417316 

267089984 
268336125 
269586136 

25.3772 
25.3969 
25.4165 

8.6357 
8.6401 
8.6446 

1 
1 
1 

.55280 
.55039 
.54799 

2023.2 
2026.3 
2029.5 

325733 
326745 
327759 

647 
648 
649 

418609 
419904 
421201 

270840023 
272097792 
273359449 

25.4362 
25.4558 
25.4755 

8.6490 
8.6535 
8.6579 

1 
1 
1 

.54560 
.54321 
.54083 

2032.6 
2035.8 
2038.9 

328775 
329792 
330810 

FUNCTIONS   OF   THE   NATURAL  NUMBERS 


No. 

Square 

Cube 

Square 
Root 

Cube 
Root 

1000  x 
Recip. 

No.—  Dia. 

Circum. 

Area 

650 
051 
652 
653 

422500 
423801 
425104 
426409 

274625000 
275894451 
277167808 
278445077 

26.4951 
25.5147 
25.5343 
25.5539 

8.6624 
8.6668 
8.6713 
8.6757 

1.53846 
1.53610 
1.53374 
1.53139 

2042.0 
2045.2 
2048.3 
2051.5 

331831 
332853 
333876 
334901 

654 
655 
656 

427716 
429025 
430336 

279726264 
281011375 
282300416 

25.5734 
25.5930 
25.6125 

8.6801 
8.6845 
8.6890 

1.52905 
1.52672 
1.52439 

2054.6 
2057.7 
2060.9 

335927 
336955 
337985 

657 
658 
659 

431649 
432964 
434281 

283593393 
284890312 
286191179 

25.6320 
25.6515 
25.6710 

8.6934 
8.6978 

8.7022 

1.52207 
1.51976 
1.51745 

2064.0 
2067.2 
2070.3 

339016 
340049 
341084 

660 

661 
662 
663 

435600 
436921 
438244 
439569 

287496000 
288804781 
290117528 
291434247 

25.6905 
25.7099 
25.7294 

25.7488 

8.7066 
8.7110 
8.7154 
8.7198 

1.51515 
1.51286 
1.51057 
1.50830 

2073.5 
2076.6 
2079.7 
2082.9 

342119 
343157 
344196 
345237 

664 
665 
666 

440896 
442225 
443556 

292754944 
294079625 
295408296 

25.7682 
25.7876 
25.8070 

8.7241 
8.7285 
8.7329 

1.50602 
1.50376 
1.50150 

2086.0 
2089.2 
2092.3 

346279 
347323 
348368 

667 
668 
669 

444889 
446224 
447561 

296740963 
298077632 
299418309 

25.8263 
25.8457 
25.8650 

8.7373 
8.7416 
8.7460 

1.49925 
1.49701 
1.49477 

2095.4 
2098.6 
2101.7 

349415 
350464 
351514 

670 

671 
672 
673 

448900 
450241 
451584 
452929 

300763000 
302111711 
303464448 
304821217 

25.8844 
25.9037 
25.9230 
25.9422 

8.7503 
8.7547 
8.7590 
8.7634 

1.49254 
1.49031 
1.48810 
1.48588 

2104.9 
2108.0 
2111.2 
2114.3 

352565 
353618 
354673 
355730 

674 
675 
676 

454276 
455625 
456976 

306182024 
307546875 
308915776 

25.9615 

25.9808 
26.0000 

8.7677 
8.7721 
8.7764 

1.48368 
1.48148 
1.47929 

2117.4 
2120.6 
2123.7 

356788 
357847 
358908 

677 
678 
679 

458329 
459684 
461041 

310288733 
311665752 
313046839 

26.0192 
26.0384 
26.0576 

8.7807 
8.7850 
8.7893 

1.47710 
1.47493 
1.47275 

2126.9 
2130.0 
2133.1 

359971 
361035 
362101 

680 

681 
682 
683 

462400 
463761 
465124 
466489 

314432000 
315821241 
317214568 
318611987 

26.0768 
26.0960 
26.1151 
26.1343 

8.7937 
8.7980 
8.8023 
8.8066 

1.47059 
1.46843 
1.46628 
1.46413 

2136.3 
2139.4 
2142.6 
2145.7 

363168 
364237 
365308 
366380 

684 
685 
686 

467856 
469225 
470596 

320013504 
321419125 
322828856 

26.1534 
26.1725 
26.1916 

8.8109 
8.8152 
8.8194 

1.46199 
1.45985 
1.45773 

2148.8 
2152.0 
2155.1 

367453 
368528 
369605 

687 
688 
689 

471969 
473344 

474721 

324242703 
325660672 
327082769 

26.2107 
26  .  2298 
26.2488 

8.8237 
8.8280 
8.8323 

1.45560 
1.45349 
1.45138 

2158.3 
2161.4 
2164.6 

370684 
371764 
372845 

690 

691 
692 
693 

476100 
477481 
478864 
480249 

328509000 
329939371 
331373888 
332812557 

26.2679 
26.2869 
26.3059 
26.3249 

8.8366 
8.8408 
8.8451 
8.8493 

1.44928 
1.44718 
1.44509 
1.44300 

2167.7 
2170.8 
2174.0 
2177.1 

373928 
375013 
376099 
377187 

694 
695 
696 

481636 
483025 
484416 

334255384 
335702375 
337153536 

26.3439 
26.3629 
26.3818 

8.8536 
8.8578 
8.8621 

1.44092 
.43885 
.43678 

2180.3 
2183.4 
2186.5 

378276 
379367 
380459 

697 
698 
699 

485809 
487204 
488601 

338608873 
340068392 
341532099 

26.4008 
26.4197 
26.4386 

8.8663 

8.8706 
8.8748 

.43472 
.43266 
.43062 

2189.7 
2192.8 
2196.0 

381553 
382649 
383746 

FUNCTIONS    OF    THE   NATURAL   NUMBERS 


No. 

Square 

Cube 

Square 
Root 

Cube 
Root 

1000  x 
Recip. 

No.=Dia. 

Circum. 

Area 

700 

701 
702 
703 

490000 
491401 
492804 
494209 

343000000 
344472101 
345948408 
347428927 

26.4575 
26.4764 
26.4953 
26.5141 

8.8790 
8.8833 
8  .  8875 
8.8917 

1.42857 
1.42653 
1.42450 
1.42248 

2199.1 
2202.3 
2205.4 
2208.5 

384845 
385945 
387047 
388151 

704 
705 
706 

495616 
497025 
498436 

348913664 
350402625 
351895816 

26.5330 
26.5518 
26.5707 

8.8959 
8.9001 
8.9043 

1.42045 
1.41844 
1.41643 

2211.7 
2214.8 
2218.0 

389256 
390363 
391471 

707 
708 
709 

499849 
501264 
502681 

353393243 
354894912 
356400829 

26.5895 
26.6083 
26.6271 

8.9085 
8.9127 
8.9169 

1.41443 
1.41243 
1.41044 

2221.1 
0094  ° 
222Y.4 

392580 
393692 
394805 

710 
711 
712 
713 

504100 
505521 
506944 
508369 

357911000 
359425431 
360944128 
362467097 

26.6458 
26.6646 
26.6833 
26.7021 

8.9211 
8.9253 
8.9295 
8.9337 

1.40845 
1.40647 
1.40449 
1.40252 

2230.5 
2233.7 
2236.8 
2240.0 

395919 
397035 
398153 
399272 

714 
715 
716 

509796 
511225 
512656 

363994344 
365525875 
367061696 

26.7208 
26.7395 

26.7582 

8.9378 
8.9420 
8.9462 

1.40056 
1.39860 
1.39665 

2243.1 
2246.2 
2249.4 

400393 
401515 
402639 

717 
718 
719 

514089 
515524 
516961 

368601813 
370146232 
371694959 

26.7769 
26.7955 
26.8142 

8.9503 
8.9545 
8.9587 

1.39470 
1.39276 
1.39082 

2252.5 
2255.7 
2258.8 

403765 
404892 
406020 

720 

721 
722 
723 

518400 
519841 
521284 
522729 

373248000 
374805361 
376367048 
377933067 

26.8328 
26.8514 
26.8701 
26.8887 

8.9628 
8.9670 
8.9711 
8.9752 

1.38889 
1.38696 
1.38504 
1.38313 

2261.9 
2265.1 
2268.2 
2271.4 

407150 
408282 
409415 
410550 

724 
725 
726 

524176 
525625 
527076 

379503424 
381078125 
382657176 

26.9072 
26.9258 
26.9444 

8.9794 
8.9835 
8.9876 

1.38122 
1.37931 
1.37741 

2274.5 

2277.7 
2280.8 

411687 
412825 
413965 

727 
728 
729 

528529 
529984 
531441 

384240583 
385828352 
387420489 

26.9629 
26.9815 
27.0000 

8.9918 
8.9959 
9.0000 

1.37552 
1.37363 
1.37174 

2283.9 
2287.1 
2290.2 

415106 
416248 
417393 

730 
731 
732 
733 

532900 
534361 
535824 
537289 

389017000 
390617891 
392223168 
393832837 

27.0185 
27.0370 
27.0555 
27.0740 

9.0041 
9.0082 
9.0123 
9.0164 

1.36986 
1.36799 
1.36612 
1.36426 

2293.4 
2296.5 
2299.6 
2302.8 

418539 
419686 
420835 
421986 

734 
735 
736 

538756 
540225 
541696 

395446904 
397065375 
398688256 

27.0924 
27.1109 
27.1293 

9.0205 
9  .  0246 

9.0287 

1.36240 
1.36054 
1.35870 

2305.9 
2309.1 
2312.2 

423138 
424293 
425447 

737 
738 
739 

543169 
544644 
546121 

400315553 
401947272 
403583419 

27.1477 
27.1662 
27.1846 

9.0328 
9.0369 
9.0410 

1.35685 
1.35501 
1.35318 

2315.4 
2318.5 
2321.6 

426604 
427762 
428922 

740 
741 
742 
743 

547600 
549081 
550564 
552049 

405224000 
406869021 
408518488 
410172407 

27.2029 
27.2213 
27.2397 
27.2580 

9.0450 
9.0491 
9.0532 
9.0572 

1.35135 
1.34953 
1.34771 
1.34590 

2324.8 
2327.9 
2331.1 
2334.2 

430084 
431247 
432412 
433578 

744 
745 
746 

553536 
555025 
556516 

411830784 
413493625 
415160936 

27.2764 
27.2947 
27.3130 

9.0613 
9.0654 
9.0694 

1.34409 
1.34228 
1.34048 

2337.3 
2340.5 
2343.6 

434746 
435916 
437087 

747 
748 
749 

558009 
559504 
561001 

416832723 
418508992 
420189749 

27.3313 

27.3496 
27.3679 

9.0735 
9.0775 
9.0816 

1.33869 
1.33690 
1.33511 

2346.8 
2349.9 
2353.1 

438259 
439433 
440609 

FUNCTIONS    OF    THE    NATURAL    NUMBERS 


67 


No. 

Square 

Cube 

Square 
Root 

Cube 
Root 

1000  x 
Recip. 

No.—  Dia. 

Circum. 

Area 

750 

751 

752 
753 

562500 
564001 
565504 
567009 

421875000 
423564751 
425259008 
426957777 

27.3861 
27.4044 
27.4226 
27.4408 

9.0856 
9.0896 
9.0937 

9.0977 

1.33333 
1.33156 
1.32979 
1.32802 

2356.2 
2359.3 
2362.5 
2365.6 

441786 
442965 
444146 
445328 

754 
755 
756 

568516 
570025 
571536 

428661064 
430368875 
432081216 

27.4591 
27.4773 
27.4955 

9.1017 
9.1057 
9.1098 

1.32626 
1.32450 
1.32275 

2368.8 
2371.9 
2375.0 

446511 
447697 
448883 

757 

758 
759 

573049 
574564 
576081 

433798093 
435519512 
437245479 

27.5136 
27.5318 
27.5500 

9.1138 
9.1178 
9.1218 

1.32100 
1.31926 
1.31752 

2378.2 
2381.3 
2384.5 

450072 
451262 
452453 

760 
761 
762 
763 

577600 
579121 
580644 
582169 

438976000 
440711081 
442450728 
444194947 

27.5681 
27.5862 
27.6043 
27.6225 

9.1258 
9.1298 
9.1338 
9.1378 

1.31579 
1.31406 
1.31234 
1.31062 

2387.6 
2390.8 
2393.9 
2397.0 

453646 
454841 
456037 
457234 

764 
765 
766 

583696 
585225 
586756 

445943744 
447697125 
449455096 

27.6405 
27.6586 
27.6767 

9.1418 
9.1458 
9.1498 

1.30890 
1.30719 
1.30548 

2400.2 
2403.3 
2406.5 

458434 
459635 
460837 

767 
768 
769 

588289 
589824 
591361 

451217663 
452984832 
454756609 

27.6948 
27.7128 
27.7308 

9.1537 
9.1577 
9.1617 

1.30378 
1.30208 
1.30039 

2409.6 
2412.7 
2415.9 

462041 
463247 
464454 

770 
771 

772 
773 

592900 
594441 
595984 
597529 

456533000 
458314011 
460099648 
461889917 

27.7489 
27.7669 
27.7849 
27.8029 

9.1657 
9.1696 
9.1736 
9.1775 

1.29870 
1.29702 
1.29534 
1.29366 

2419.0 

2422.2 
2425.3 
2428.5 

465663 
466873 
468085 
469298 

774 
775 
776 

599076 
600625 
602176 

463684824 
465484375 
467288576 

27.8209 
27.8388 
27.8568 

9.1815 
9.1855 
9.1894 

1.29199 
1.29032 
1.28866 

2431.6 
2434.7 
2437.9 

470513 
471730 
472948 

777 
778 
779 

603729 
605284 
606841 

469097433 
470910952 
472729139 

27.8747 
27.8927 
27.9106 

9.1933 
9.1973 
9.2012 

1.28700 
1.28535 
1.28370 

2441.0 
2444.2 
2447.3 

474168 
475389 
476612 

780 

781 

782 
783 

608400 
609961 
611524 
613089 

474552000 
476379541 
478211768 
480048687 

27.9285 
27.9464 
27.9643 
27.9821 

9.2052 
9.2091 
9.2130 
9.2170 

1.28205 
1.28041 
1.27877 
1.27714 

2450.4 
2453.6 
2456.7 
2459.9 

477836 
479062 
480290 
481510 

784 
785 
786 

614656 
616225 
617796 

481890304 
483736625 

485587656 

28.0000 
28.0179 
28.0357 

9.2209 
9.2248 

9.2287 

1.27551 
1.27389 
1.27226 

2463.0 
2466.2 
2469.3 

482750 
483982 
485216 

787 
788 
789 

619369 
620944 
622521 

487443403 
489303872 
491169069 

28.0535 
28.0713 
28.0891 

9  2326 
9.2365 
9.2404 

1.27065 
1.26904 
1.26743 

2472.4 
2475.6 

2478.7 

486451 
487688 
488927 

790 
791 
792 
793 

624100 
625681 
627264 
628849 

493039000 
494913671 
496793088 
498677257 

28.1069 
28.1247 
28.1425 
28.1603 

9.2443 
9.2482 
9.2521 
9.2560 

1.26582 
1.26422 
1.26263 
1.26103 

2481.9 
2485.0 
2488.1 
2491.3 

490167 
49140D 
492652 
493897 

794 
795 
796 

630436 
632025 
633616 

500566184 
502459875 
504358336 

28.1780 
28.1957 
28.2135 

9.2599 
9.2638 
9.2677 

1.25945 
1.25786 
1.25628 

2494.4 
2497.6 
2500.7 

495143 
496391 
497641 

797 
798 
799 

635209 
636804 
638401 

506261573 
508169592 
510082399 

28.2312 
28.2489 
28.2666 

9.2716 
9.2754 
9.2793 

1.25471 
1.25313 
1.25156 

2503.8 
2507.0 
2510.1 

498892 
500145 
501399 

68 


FUNCTIONS    OF    THE    NATURAL    NUMBERS 


No. 

Square 

Cube 

Square 
Root 

Cube 
Root 

1000  x 
Recip. 

No.—  Dia. 

Circum. 

Area 

800 

801 
802 
803 

640000 
641601 
643204 
644809 

512000000 
513922401 
515849608 
517781627 

28.2843 
28.3019 
28.3196 
28.3373 

9.2832 
9.2870 
9.2909 
9.2948 

1.25000 
1.24844 
1  .  24688 
1.24533 

2513.3 
2516.4 
2519.6 
2522.7 

502655 
503912 
505171 
506432 

804 
805 

806 

646416 
648025 
649636 

519718464 
521660125 
523606616 

28.3549 
28.3725 
28.3901 

9.2986 
9.3025 
9.3063 

1.24378 
1.24224 
1.24069 

2525.8 
2529.0 
2532.1 

507694 
508958 
510223 

807 
808 
809 

651249 
652864 
654481 

525557943 
527514112 
529475129 

28.4077 
28.4253 
28.4429 

9.3102 
9.3140 
9.3179 

1.23916 
1.23762 
1.23609 

2535.3 
2538.4 
2541.5 

511490 
512758 
514028 

810 
811 
812 
813 

656100 
657721 
659344 
660969 

531441000 
533411731 
535387328 
537367797 

28.4605 

28.4781 
28.4956 
28.5132 

9.3217 
9.3255 
9.3294 
9.3332 

1.23457 
1.23305 
1.23153 
1.23001 

2544.7 
2547.8 
2551.0 
2554.1 

515300 
516573 
517848 
519124 

814 
815 
816 

662596 
664225 
665856 

539353144 
541343375 
543338496 

28.5307 
28.5482 
28.5657 

9.3370 
9.3408 
9.3447 

1.22850 
1.22699 
1.22549 

2557.3 
2560.4 
2563.5 

520402 
521681 
522962 

817 
818 
819 

667489 
669124 
670761 

545338513 
547343432 
549353259 

28.5832 
28.6007 
28.6182 

9.3485 
9.3523 
9.3561 

1.22399 
1  22249 
1.22100 

2566.7 
2569.8 
2573.0 

524245 
525529 
526814 

820 
821 
822 
823 

672400 
674041 
675684 
677329 

551368000 
553387661 
555412248 
557441767 

28.6356 
28.6531 
28.6705 
28.6880 

9.3599 
9.3637 
9.3675 
9.3713 

1.21951 
1.21803 
1.21655 
1.21507 

2576.1 
2579.2 
2582.4 
2585.5 

528102 
529391 
530681 
531973 

824 
825 
826 

678976 
680625 
682276 

559476224 
561515625 
563559976 

28.7054 

28.7228 
28.7402 

9.3751 
9.3789 
9.3827 

1.21359 
1.21212 
1.21065 

2588.7 
2591.8 
2595.0 

533267 
534562 
535858 

827 
828 
829 

683929 
685584 
687241 

565609283 
567663552 
569722789 

28.7576 
28.7750 
28.7924 

9.3865 
9.3902 
9.3940 

1.20919 
1.20773 
1.20627 

2598.1 
2601.2 
2604.4 

537157 
538456 
539758 

830 

831 
832 
833 

688900 
690561 
692224 
693889 

571787000 
573856191 
575930368 
578009537 

28.8097 
28.8271 
28.8444 
28.8617 

9.3978 
9.4016 
9.4053 
9.4091 

1.20482 
1.20337 
1.20192 
1.20048 

2607.5 
2610.7 
2613.8 
2616.9 

541061 
542365 
543671 
544979 

834 
835 
836 

695556 
697225 
698896 

580093704 
582182875 
584277056 

28.8791 
28.8964 
28.9137 

9.4129 
9.4166 
9.4204 

1.19904 
1.19760 
1.19617 

2620.1 
2623.2 
2626.4 

546288 
547599 
548912 

837 
838 
839 

700569 
702244 
703921 

586376253 
588480472 
590589719 

28.9310 

28.9482 
28.9655 

9.4241 
9.4279 
9.4316 

1.19474 
1.19332 
1.19190 

2629.5 
2632.7 
2635.8 

550226 
551541 

552858 

840 

841 
842 

843 

705600 
707281 
708964 
710649 

592704000 
594823321 
596947688 
599077107 

28.9828 
29.0000 
29.0172 
29.0345 

9.4354 
9.4391 
9.4429 
9.4466 

1.19048 
1.18906 
1.18765 
1.18624 

2638.9 
2642.1 
2645.2 
2648.4 

554177 
555497 
556819 
558142 

844 
845 
846 

712336 
714025 
715716 

601211584 
603351125 
605495736 

29.0517 
29.0689 
29.0861 

9.4503 
9.4541 
9.4578 

1.18483 
1.18343 
1.18203 

2651.5 
2654.6 
2657.8 

559467 
560794 
562122 

847 
848 
849 

717409 
719104 
720801 

607645423 
609800192 
611960049 

29.1033 
29.1204 
29.1376 

9.4615 
9.4652 
9.4690 

1.18064 
1.17925 
1.17786 

2660.9 
2664.1 
2667.2 

563452 
564783 
566116 

FUNCTIONS    OF    THE    NATURAL    NUMBERS 


No. 

Square 

Cube 

Square 
Root 

Cube 
Root 

1000  x 
Recip. 

No.^Dia. 

Circum. 

Area 

850 

851 
852 
853 

722500 
724201 
725904 
727609 

614125000 
616295051 
618470208 
620650477 

29.1548 
29.1719 
29.1890 
29.2062 

9.4727 
9.4764 
9.4801 
9.4838 

1 
1 
1 

1 

.17647 
.17509 
.17371 
.17233 

2670.4 
2673.5 
2676.6 
2679.8 

567450 
568786 
570124 
571463 

854 
855 
856 

729316 
731025 
732736 

622835864 
625026375 
627222016 

29.2233 
29.2404 
29.2575 

9.4875 
9.4912 
9.4949 

1 

1 
1 

.17096 
.16959 
.16822 

2682  9 
2686.1 
2689.2 

572803 
574146 
575490 

857 
858 
859 

734449 
736164 
737881 

629422793 
631628712 
633839779 

29.2746 
29.2916 
29.3087 

9.4986 
9.5023 
9.5060 

1 

1 

1 

.16686 
.16550 
.16414 

2692.3 
2695.5 
2698.6 

576835 
578182 
579530 

860 

861 
862 
863 

739600 
741321 
743044 
744769 

636056000 
638277381 
640503928 
642735647 

29.3258 
29.3428 
29.3598 
29.3769 

9.5097 
9.5134 
9.5171 
9.5207 

1 

1 
1 
1 

.16279 
.16144 
.16009 
.15875 

2701.8 
2704.9 
2708.1 
2711.2 

580880 
582232 
583585 
584940 

864 
£65 

866 

746496 
748225 
749956 

644972544 
647214625 
649461896 

29.3939 
29.4109 
29.4279 

9.5244 
9.5281 
9.5317 

1 

1 
1 

.15741 
.15607 
.15473 

2714.3 
2717.5 
2720.6 

586297 
587655 
589014 

867 
868 
869 

751689 
753424 
755161 

651714363 
653972032 
656234909 

29.4449 
29.4618 
29.4788 

9.5354 
9.5391 
9.5427 

1 
1 
1 

.15340 
.15207 
.15075 

2723.8 
2726.9 
2730.0 

590375 
591738 
593102 

870 

871 
872 
873 

756900 
758641 
760384 
762129 

658503000 
660776311 
663054848 
665338617 

29.4958 
29.5127 
29.5296 
29.5466 

9.5464 
9.5501 
9.5537 
9.5574 

1 
1 

1 
1 

.14943 
.14811 
.14679 
.14548 

2733.2 
2736.3 
2739.5 
2742.6 

594468 
595835 
597204 
598575 

874 
875 

876 

763876 
765625 
767376 

667627624 
669921875 
672221376 

29.5635 
29.5804 
29.5973 

9.5610 
9.5647 
9.5683 

1 
1 
I 

.14416 
.14286 
.14155 

2745.8 
2748.9 
2752.0 

599947 
601320 
602696 

877 
878 
879 

769129 
770884 
772641 

674526133 
676836152 
679151439 

29.6142 
29.6311 
29.6479 

9.5719 
9.5756 
9.5792 

1 

1 
1 

.14025 
.13895 
.13766 

2755.2 
2758.3 
2761.5 

604073 
605451 
606831 

880 

881 
882 
883 

774400 
7"76161 
777924 
779689 

681472000 
683797841 
686128968 
688465387 

29.6648 
29.6816 
29.6985 
29.7153 

9.5828 
9.5865 
9.5901 
9.5937 

1 
1 
1 
1 

.13636 
.13507 
.13379 
.13250 

2764.6 
2767.7 
2770.9 
2774.0 

608212 
609595 
610980 
612366 

884 
885 
886 

781456 
783225 
784996 

690807104 
693154125 
695506456 

29.7321 
29.7489 
29.7658 

9.5973 
9.6010 
9.6046 

1 
1 
1 

.13122 
.12994 
.12867 

2777.2 
2780.3 
2783.5 

613754 
615143 
616534 

887 
888 
889 

786769 
788544 
790321 

697864103 
700227072 
702595369 

29.7825 
29.7993 
29.8161 

9.6082 
9.6118 
9.6154 

1 
1 
1 

.12740 
.12613 
.12486 

2786.6 
2789.7 
2792.9 

617927 
619321 
620717 

890 
891 
£92 
893 

792100 
793881 
795664 
797449 

704969000 
707347971 
709732288 
712121957 

29.8329 
29.8496 
29.8664 
29.8831 

9.6190 
9.6226 
9.6262 
9.6298 

1 
1 
1 
I 

.12360 
.12233 
.12108 
.11982 

2796.0 
2799.2 
2802.3 
2805.4 

622114 
623513 
624913 
626315 

894 
895 

896 

799236 
801025 
802816 

714516984 
716917375 
719323136 

29.8998 
29.9166 
29.9333 

9.6334 
9.6370 
9.6406 

I 
1 
1 

.11857 
.11732 
.11607 

2808.6 
2811.7 
2814.9 

627718 
629124 
630530 

897 
898 
899 

804609 
806404 
808201 

721734273 

724150792 
726572699 

29.9500 
29.9666 
29.9833 

9.6442 
9.6477 
9.6513 

1 

1 
] 

.11483 
.11359 
.11235 

2818.0 
2821.2 
2824.3 

631938 
633348 
634760 

FUNCTIONS    OF    THE    NATURAL    NUMBERS 


No. 

Square 

Cube 

Square 
Root 

Cube 
Root 

1000  x 
Recip. 

No.=Dia. 

Circum. 

Area 

900 

901 
902 
903 

810000 
811801 
813604 
815409 

729000000 
731432701 
733870808 
736314327 

30.0000 
30.0167 
30.0333 
30.0500 

9.6549 
9.6585 
9.6620 
9.6656 

1.11111 

1.10988 
1.10865 
1.10742 

2827.4 
2830.6 
2833.7 
2836.9 

636173 
637587 
639003 
640421 

904 
905 
906 

817216 
819025 
820836 

738763264 
741217625 
743677416 

30.0666 
30.0832 
30.0998 

9.6692 
9.6727 
9.6763 

1.10619 
1.10497 
1.10375 

2840.0 
2843.1 
2846.3 

641840 
643261 
644683 

907 
908 
909 

822649 
824464 
826281 

746142643 
748613312 
751089429 

30.1164 
30.1330 
30.1496 

9.6799 
9.6834 
9.6870 

1.10254 
1.10132 
1.10011 

2849.4 
2852.6 
2855.7 

646107 
647533 
648960 

910 
911 
912 
913 

828100 
829921 
831744 
833569 

753571000 
756058031 
758550528 
761048497 

30.1662 
30.1828 
30.1993 
30.2159 

9.6905 
9.6941 
9.6976 
9.7012 

1.09890 
1.09769 
1.09649 
1.09529 

2858.8 
2862.0 
2865.1 
2868.3 

650388 
651818 
653250 
654684 

914 
915 
916 

835396 
837225 
839056 

763551944 
766060875 
768575296 

30.2324 
30.2490 
30.2655 

9.7047 
9.7082 
9.7118 

1.09409 
1.09290 
1.09170 

2871.4 
2874.6 

2877.7 

656118 
657555 
658993 

917 
918 
919 

840889 
842724 
844561 

771095213 
773620632 
776151559 

30.2820 
30.2985 
30.3150 

9.7153 
9.7188 
9.7224 

1.09051 
1.08932 
1.08814 

2880.8 
2884.0 
2887.1 

660433 
661874 
663317 

920 

921 
922 
923 

846400 
848241 
850084 
851929 

778688000 
781229961 
783777448 
786330467 

30.3315 
30.3480 
30.3645 
30.3809 

9.7259 
9.7294 
9.7329 
9.7364 

1.08696 
1.08578 
1.08460 
1.08342 

2890.3 
2893  .  4 
2896.5 
2899.7 

664761 
666207 
667654 
669103 

924 
925 
926 

853776 
855625 
857476 

788889024 
791453125 
794022776 

30.3974 
30.4138 
30.4302 

9.7400 
9.7435 
9.7470 

1.08225 
1.08108 
1.07991 

2902.8 
2906.0 
2909.1 

670554 
672006 
673460 

927 
928 
929 

859329 
861184 
863041 

796597983 
799178752 
801765089 

30.4467 
30.4631 
30.4795 

9.7505 
9.7540 
9.7575 

1.07875 
1.07759 
1.07643 

2912.3 
2915.4 
2918.5 

674915 
676372 
677831 

930 
931 
932 
933 

864900 
866761 
868624 
870489 

804357000 
806954491 
809557568 
812166237 

30.4959 
30.5123 
30.5287 
30.5450 

9.7610 
9.7645 
9.7680 
9.7715 

1.07527 
1.07411 
1.07296 
1.07181 

2921.7 
2924.8 
2928.0 
2931.1 

679291 
680752 
682216 
683680 

934 
935 
936 

872356 
874225 
876096 

814780504 
817400375 
820025856 

30.5614 
30.5778 
30.5941 

9.7750 
9.7785 
9.7819 

1.07066 
1.06952 
1.06838 

2934.2 
2937.4 
2940.5 

685147 
686615 
688084 

937 
938 
939 

877969 
879844 
881721 

822656953 
825293672 
827936019 

30.6105 
30.6268 
30.6-131 

9.7854 
9.7889 
9.7924 

1.06724 
1.06610 
1.06496 

2943.7 

2946.8 
2950.0 

689555 
691028 
692502 

940 

941 
942 
943 

883600 
885481 
887364 
889249 

830584000 
833237621 
835896888 
838561807 

30.6594 
30.6757 
30.6920 
30.7083 

9.7959 
9.7993 
9.8028 
9.8063 

1.06383 
1.06270 
1.06157 
1.06045 

2953.1 
2956.2 
2959.4 
2962.5 

693978 
695455 
696934 
698415 

944 
945 
946 

891136 
893025 
894916 

841232384 
843908625 
846590536 

30.7246 
30.7409 
30.7571 

9.8097 
9.8132 
9.8167 

1.05932 
1.05820 
1.05708 

2965.7 
2968.8 
2971.9 

699897 
701380 
702865 

947 
948 
949 

896809 
898704 
900601 

849278123 
851971392 
854670349 

30.7734 
30.7896 
30.8058 

9.8201 
9.8236 

9.8270 

1.05597 
1.05485 
1.05374 

2975.1 

2978.2 
2981.4 

70435? 
705840 
707330 

FUNCTIONS    OF    THE    NATURAL    NUMBERS 


71 


No. 

Square 

Cube 

Square 
Root 

Cube 
Root 

1000  x 
Recip. 

No.=rDia. 

Circum. 

Area 

950 
951 
952 
953 

902500 
904401 
906304 
908209 

857375000 
860085351 
862801408 
865523177 

30.8221 
30.S383 
30.8545 
30.8707 

9.S305 
9.8339 
9.8374 
9.8408 

1.05263 
1.05152 
1.05042 
1.04932 

2984.5 
2987.7 
2990.8 
2993.9 

708822 
710315 
711809 
713306 

954 
955 
956 

910116 
912025 
913936 

868250664 
870983875 
873722816 

30.8869 
30.9031 
30.9192 

9.8443 
9.8477 
9.8511 

1.04822 
1.04712 
1.04603 

2997.1 
3000.2 
3003.4 

714803 
716303 
717804 

957 
958 
959 

915849 
917764 
919681 

876467493 
879217912 
881974079 

30.9354 
30.9516 
30.9677 

9.8546 
9.8580 
9.8614 

1.04493 
1.04384 
1.04275 

3006.5 
3009.6 
3012.8 

719306 
720810 
722316 

960 
961 
962 
963 

921600 
923521 
925444 
927369 

884736000 
887503681 
890277128 
893056347 

30.9839 
31.0000 
31.0161 
31.0322 

9.8648 
9.8683 
9.8717 
9.8751 

1.04167 
1.04058 
1.03950 
1.03842 

3015.9 
3019.1 
3022:2 
3025.4 

723823 
725332 
726842 
728354 

964 
965 
966 

929296 
931225 
933156 

895841344 
898632125 
901428696 

31.0483 
31.0644 
31.0805 

9.8785 
9.8819 
9.8854 

1.03734 
1.03627 
1.03520 

3028.5 
3031.6 
3034.8 

729867 
731382 
732899 

967 
968 
969 

935089 
937024 
938961 

904231063 
907039232 
909853209 

31.0966 
31.1127 
31.1288 

9.8888 
9.8922 
9.8956 

1.03413 
1.03306 
1.03199 

3037.9 
3041.1 
3044.2 

734417 
735937 
737458 

970 
971 
972 
973 

940900 
942841 
944784 
946729 

912673000 
915498611 
918330048 
921167317 

31.1448 
31.1609 
31.1769 
31.1929 

9.8990 
9.9024 
9.9058 
9.9092 

1.03093 
1.02987 
1.02881 
1.02775 

3047.3 
3050.5 
3053  .  6 
3056.8 

738981 
740506 
742032 
743559 

974 
975 
976 

948676 
950625 
952576 

924010424 
926859375 
929714176 

31.2090 
31.2250 
31.2410 

9.9126 
9.9160 
9.9194 

1.02669 
1.02564 
1.02459 

3059.9 
3063.1 
3066.2 

745088 
746619 
748151 

977 
978 
979 

954529 
956484 
958441 

932574833 
935441352 
938313739 

31.2570 
31.2730 
31.2890 

9.9227 
9.9261 
9.9295 

1.02354 
1.02249 
1.02145 

3069.3 
3072.5 
3075.6 

749685 
751221 
752758 

980 
981 
982 
983 

960400 
962361 
964324 
966289 

941192000 
944076141 
946966168 
949862087 

31.3050 
31.3209 
31.3369 
31.3528 

9.9329 
9.9363 
9.9396 
9.9430 

1.02041 
1.01937 
1.01883 
1.01729 

3078.8 
3081.9 
3085.0 
3088.2 

754296 
755837 
757378 

758922 

984 
985 
986 

968256 
970225 
972196 

952763904 
955671625 
958585256 

31  .3688 
31.3847 
31.4006 

9.9464 
9.9497 
9.9531 

1.01626 
1.01523 
1.01420 

3091.3 
3094.5 
3097.6 

760466 
762013 
763561 

987 
988 
989 

974169 
976144 
978121 

961504803 
964430272 
967361669 

31.4166 
31.4325 
31.4484 

9.9565 
9.9598 
9.9632 

1.01317 
1.01215 
1.01112 

3100.8 
3103.9 
3107.0 

765111 
766662 
768214 

990 
£91 
992 
993 

980100 
982081 
984064 
986049 

970299000 
973242271 
976191488 
979146657 

31.4643 
31.4802 
31.4960 
31.5119 

9.9666 
9.9699 
9.9733 
9.9766 

1.01010 
1.00908 
1.00806 
1.00705 

3110.2 
3113.3 
3116.5 
3119.6 

769769 
771325 
772882 
774441 

994 
995 
996 

988036 
990025 
992016 

982107784 
985074875 
988047936 

31.5278 
31.5436 
31.5595 

9.9800 
9.9833 
9.9866 

1.00604 
1.00503 
1.00402 

3122.7 
3125.9 
3129.0 

776002 
777564 
779128 

997 
998 
999 

994009 
996004 
998001 

991026973 
994011992 
997002999 

31.5753 
31.5911 
31.6070 

9.9900 
9.9933 
9.9967 

1.00301 
1.00200 
1.00100 

3132.2 
3135.3 
3138.5 

780693 
782260 
78382k 

72  FUNCTIONS    OF    THE    NATURAL    NUMBERS 

Fifth   Roots  and  Fifth   Powers 


Root 

Power 

Root 

Power 

Root 

Power 

1 

2 

3 

1 
32 
243 

34 
35 

36 

45435424 
52521875 
60466176 

67 
68 
69 

1350125101 
145393356S 
1564031349 

4 
5 
6 

1024 
3125 

7776 

37 
38 
39 

69343957 
79235168 
90224199 

70 
71 

72 

1680700000 
1804229351 
1934917632 

7 
8 
9 

16807 
32768 
59049 

40 
41 
42 

102400000 
115856201 
130691231 

73 

74 

75 

2073071593 
2219006624 
2373046875 

10 
11 

ia 

100000 
161051 
248832 

43 
44 
45 

147008443 
164916224 
184528125 

76 

77 
78 

2535525376 
•  2706784157 
2887174368 

13 
14 
15 

371293 
537824 
759375 

46 
47 

48 

205962976 
229345007 
254803968 

79 
80 
81 

3077056399 
3276800000 
3486784401 

16 

17 
18 

1048576 
1419857 
.  1889568 

49 
50 
51 

282475249 
312500000 
345025251 

82 
83 
84 

3707398432 
3939040643 
4182119424 

19 

20 
21 

2476099 
3200000 
4084101 

52 
53 
54 

380204032 
418195493 
459165024 

85 
86 

87 

4437053125 
4704270176 
4984209207 

22 
23 
24 

5153632 
6436343 
7962624 

55 
56 
57 

503284375 
550731776 
601692057 

88 
89 
90 

5277319168 
5584059449 
5904900000 

25 
26 

27 

9765625 

11881376 
14348907 

58 
59 
60 

656356768 
714924299 
777600000 

91 
92 
93 

6240321451 
6590815232 
6956883693 

28 
29 
30 

17210368 
20511149 
24300000 

61 
62 
63 

844596301 
916132832 
992436543 

94 
95 
96 

7339040224 
7737809375 
8153726976 

31 
32 
33 

28629151 
33554432 
39135393 

64 
65 
66 

1073741824 
1160290625 
1252332576 

97 
98 
99 

8587340257 
903920796S 
9509900499 

Square  Roots  and 

Cube  Roots  of 

Fractions 

Fract- 

Sq.  rt.   Cu.  rt. 

Fract. 

Sq.  rt.   Cu.  rt. 

Fract. 

Sq.  rt.   Cu.  rt. 

1-3 
2-3 
1-4 

3-4 

.57735   .69336 
.81650   .87358 
.50000   .62996 
.86603   .90856 

3-7 
4-7 
5-7 
6-7 

.65465   .75395 
.75593   .82983 
.84515   .89390 
.92582   .94991 

1-9 
2-9 
4-9 
5-9 

.33333   .48075 
.47140   .60571 
.66667   .76314 
.74536   .82207 

1-6 
5-6 
1-7 
2-7 

.40825   .55032 
.91287   .94104 
.37796   .52276 
.53452   .65863 

1-8 
3-8 
5-8 
7-8 

.35355   .50000 
.61237   .72112 
.79057   .85449 
.93541   .95647 

7-9 
1-12 
5-12 
7-12 

.88192   .91964 
.28868   .43679 
.64550   .74690 
.76376   .83555 

CONVERSION   OF   UNITS 

CONVERSION  OF  UNITS 
Decimal   Equivalents  of   Fractions  of  One   Inch. 


Fract-   Dec. 

Fract.   Dec.     Fract.   Dec. 

Fract. 

Dec. 

1-64   .015625     17-64   .265625    33-64  .515625 
1-32   .03125       9-32   .28125     17-32   .53125 
3-64  .046875     19-64  .296875     35-64  .546875 

1-16  .0625       5-16  .3125       9-16  .5625 
5-64   .078125     21-64   .328125    37-64   .578125 
3-32  .09375      11-32  .34375     19-32  .59375 
7-64  .109375     23-64  .359375     39-64  .609375 

1-8   .125        3-8   .375        5-8   .625 
9-64  .140625     25-64  .390625    41-64  .640625 
5-32   .15625      13-32   .40625     21-32   .65625 
11-64  .171875     27-64  .421875    43-64  .671875 

3-16  .1875       7-16  .4375      11-16  .6875 
13-64  .203125     29-64   .453125    45-64   .703125 
7-32   .21875      15-32   .46875     23-32   .71875 
15-64  .234375     31-64  .484375    47-64  .734375 
1-4   .25         1-2   .5         3-4   .75 

49-64 
25-32 
51-64 

13-16 
53-64 
27-32 
55-64 

7-8 

57-64 
29-32 
59-64 

15-16 

61-64 
31-32 
63-64 

.765625 

.78125 
.796875 

.8125 
.828125 
.84375 
.859375 

.875 
.890625 
.90625 
.921875 

.9375 
.953125 
.96875 
.984375 

Inches  in  Decimal  Parts  of  i 

i  Foot. 

In. 

0 

1-16 

1-8 

3-16 

1-4 

5-16 

3-8 

7-16 

0 

1 

2 

0 
.0833 
.1667 

.00521 
.0885 
.1719 

.01042 
.0938 
.1771 

.01562 
.0990 
.1823 

.02083 
.1042 
.1875 

.02604 
.1094 
.1927 

.03125 
.1146 
.1979 

.03646 
.1198 
.2031 

3 
4 
5 

.2500 
.3333 
.4167 

.2552 
.3385 
.4219 

.2604 
.3438 
.4271 

.2656 
.3490 
.4323 

.2708 
.3542 
.4375 

.2760 
.3594 
.4427 

.2813 
.3646 
.4479 

.2865 
.3698 
.4531 

6 

7 
8 

.5000 
.5833 
.6667 

.5052 
.5885 
.6719 

.5104 
.5938 
.6771 

.5156 

.5990 
.6823 

.5208 
.6042 

.6875 

.5260 
.6094 
.6927 

.5313 
.6146 
.6979 

.5365 
.6198 
.7031 

9 
10 
11 

.7500 
.8333 
.9167 

.7552 
.8385 
.9219 

.7604 
.8438 
.9271 

.7656 
.8490 
.9323 

.7708 
.8542 
.9375 

.7760 
.8594 
.9427 

.7813 
.8646 
.9479 

.7865 
.8698 
.9531 

In. 

1-2 

9-16 

5-8 

11-16 

3-4 

13-16 

7-8 

15-16 

0 
1 
2 

.04167 
.1250 
.2083 

.04687 
.1302 
.2135 

.05208 
.1354 
.2188 

.05729 
.1406 
.2240 

.06250 
.1458 
.2292 

.06771 
.1510 
.2344 

.07292 
.1563 
.2396 

.07812 
.1615 

.2448 

3 
4 
5 

.2917 
.3750 
.4583 

.2969 
.3802 
.4635 

.3021 
.3854 
.4688 

.3073 
.3906 
.4740 

.3125 
.3958 
.4792 

.3177 
.4010 
.4844 

.3229 
.4063 
.4896 

.3281 
.4115 
.4948 

6 

7 
8 

.5417 
.6250 
.7083 

.5469 
.6302 
.7135 

.5521 
.6354 
.7188 

.5573 
.6406 
.7240 

.5625 
.6458 
.7292 

.5677 
.6510 
.7344 

.5729 
.6563 
.7396 

.5781 
.6615 
.7448 

9 
10 
11 

.7917 
.8750 
.9583 

.7969 
.8802 
.9635 

.8021 

.8854 
.9688 

.8073 
.8906 
.9740 

.8125 
.8958 
.9792 

.8177 
.9010 
.9844 

.8229 
.9063 
.9896 

.8281 
.9115 
.9948 

74  CONVERSION   OF   UNITS 

U.  S.  AND   BRITISH    UNITS 

Lengths.     U.  S.  and  British. 

mil  =  .001  inch. 

foot  =  12  inches. 

yard  =  3  feet  =  36  inches. 

rod,  pole,  or  perch  =  5%  yards  =16^  feet. 

furlong  =  40  rods  =  220  yards  =  660  feet. 
1   statute  mile  =  8    furlongs=320    rods  =1760    yards  =  5280    feet. 
1  link=7.92  inches. 
1   chain=100  links=66  feet=4  rods. 

1  fathom=6  feet=8  spans=72  inches=  Yi2o  cable's  length. 
1   nautical    mile    (U.    S.    Coast    and    Geodetic    Survey)=1.1516 
statute  miles=6080.26  feet. 

Areas.     U.  S.  and  British. 

1  circular  mil=area  circle  1  mil  diameter=. 0000007854  square 

inches. 

1   square  foot=144  square  inches.      100  square  feet=l  square. 
1   square  yard=9  square  feet=1296  square  inches. 
1  acre=  10  square  chains=4  roods=160  square  rods 

=4840  square  yards=43560  square  feet. 
1  section=l  mile  square=640  acres. 


Volumes  and  Capacity.     U.  S. 

1  cubic  foot=1728  cubic  inches. 

1  cubic  yard=27  cubic  feet. 

1  perch  of  masonry=24.75  cubic  feet=  16 ^  feetXlK  feetXl  foot 

1  gallon  (U.  S.  liquid)=4  quarts  =  8  pints  =  231  cubic  inches. 

1  bushel  =  8  gallons  (Dry  meas.)=2150.42  cubic  inches 

=  1.24445  cubic  feet. 
1  barrel  =  31.5  gallons  =  4.2 11  cubic  feet. 


Weights.     U.  S.  and  British. 

Troy.      1   pound  =  12  ounces  =  240  pennyweight  =  5760  grains. 
Avoirdupois.      1  pound  =16  ounces  =  256  drams  =7000  grains. 

1    ton    (gross)  =  20    hundredweight    (gross)  =2240 
pounds. 

1  ton  (net)  =  20  hundredweight  =  2000  pounds. 


CONVERSION   OF   UNITS  75 


METRIC    UNITS 

Length. 

1  meter  =  10  decimeters  =100  centimeters  =  1000  millimeters. 
1  kilometer  =10  hectometers  =  100  decameters  =  1000  meters. 

Area. 

1  square  meter  or  centiare  =  100  square  decimeters  =10000  square 
centimeters. 

1  hectare  =  100  ares  =10000  centiares. 

I 

Volume  and  Capacity. 

1  cubic  meter,  or  stere=1000  cubic  decimeters 

=  1,000,000  cubic  centimeters. 

1  liter=l  cubic  decimeter  =  10  deciliters  =  100  centiliters. 
1  kiloliter,  or  stere  =  10  hectoliters  =  100  decaliters  =  1000  liters 

Weights. 

1  gram  =10  decigrams  =  100  centigrams. 

1  kilogram  =  10  hectograms  =  100  decagrams  =1000  grams. 

1  tonne,  or  metric  ton  =10  quintals  =  1000  kilograms. 

Note — Abbreviations    for  the  English  units  are  those  in  common 
use.     The  following  are  used  for  the  metric: 

mm millimeters     km kilometer     mg milligram 

cm centimeter     hect hectare     eg centigram 

m meter     g gram     kg kilogram 

Gallons  are  U.  S.  liquid  measure.     Pounds  are  Avoirdupois. 
Calorie  is  gram-centigrade  unit. 

In  the  logarithm  column,  negative  characteristics  have  been 
increased  by  10  and  are  shown  in  black  type. 


76  CONVERSION    OF   UNITS 

METRIC    TO    U.    S.    AND    U.    S.    TO    METRIC 

Lengths 


No. 

Mm.  to  64th 
of  In. 

Cm.  to 
In. 

M.  to 

Ft. 

Km.  to 

Miles 

64th  of  In. 
to  Mm. 

In.  to 

Cm. 

Ft.  to 
M. 

Miles  to 
Km. 

j 

9 

5197 

0  3937 

3 

2808 

0.62137 

0.3969 

2.54 

0. 

3048 

1.60935 

o 

5   0394 

0.7874 

6 

5617 

1.24274 

0.7938 

5.08 

0.6096 

3.21869 

3 

7 

.5590 

1.1811 

9 

.8425 

1.86411 

1.1906 

7.62 

0. 

9144 

4.82804 

10 

0787 

1.5748 

13 

1233 

2.48548 

1.5875 

10.16 

1. 

2192 

6.43739 

>-, 

12 

.5984 

1.9685 

16 

4042 

3.10685 

1.9844 

12.70 

1. 

5240 

8.04674 

6 

15 

.1181 

2.3622 

19, 

,6850 

3.72822 

2.3813 

15.24 

1 

,8288 

9.65608 

7 

ft 

17 
20 

.6378 
.1574 

2.7559 
3.1496 

22 
26 

,9658 
2467 

4.34959 
4.97096 

2.7781 
3.1750 

17.78 
20.32 

2 

1336 

4384 

11.26543 

12.87478 

9 

22 

.6771 

3.5433 

29 

,5275 

5.59233 

3.5719 

22.86 

2 

,74:i2 

14.48412 

Areas 


Sq.Cm- 

Sq.  M. 

Sq.  M. 

Hect.      Sq.  Km.       Sq.  In.      Sq.  ft.      Sq.  Y.     Acres  Sq.  Mile 

No.    to 

to 

to 

to             to                 to              to              to           to             to 

Sq.  In. 

Sq.  Ft- 

Sq.  Y. 

Acres     Sq.  Mile     Sq.  Cm.     Sq.  M.      Sq.  M.  Hect.     Sq.  Km. 

1     0.155 
2     0.310 
3     0.465 

10.7639 
21.5277 
32.2916 

1.196 
2.392 
3.588 

2.471    0.3861     6.4516   0.0929   0.8361   0.4047     2.59 
4  942    0  7722    12.9032    0.1858    1.6723    0.8094     5.18 
7  413    1.1583    19.3549    0.2787   2.5084   1.2141     7.77 

4  0  620  43  0555  4.784  9.884  1.5444  25.8065  0.3716  3.3445  1.6188  10.36 

5  0  775  53  8193  5.980  12.355  1.9305  32.2581  0.4645  4.1806  2.0235  12.95 

6  0^30  64:5832  7.176  14.826  2.3166  38.7097  0.5574  5.0168  2.4282  15.54 

7  1  085  75.3471  8.372  17.297  2.7027  45.1614  0.6503  5,8529  2.8329  18.13 

8  1240  86.1109  9.568  19.768  3.0888  51.6130  0.7432  6.6890  3.2376  20.72 

9  1  395  96  8748  10.764  22.239  3.4748  58.0646  0.8361  7.5252  3.6423  23.31 


Volumes 


Cu.  Cm.  Cu.  M.     Cu.  M.     Litres 
No.     to  to  to  to 

Cu.  In.     Cu.  Ft.   Cu.  Yds.     Gal. 


Cu.  In.     Cu.  Ft.  Cu.  Yds.      Gal-         Gal. 
to  to  to  to  to 

Cu.  M.     Cu.  M-     Litres     Cu.  M. 


1  .0610  35.314 

2  .1220  70.629 

3  .1831  105.943 


1.308 
2.616 
3.924 


.2642  264.17 
.5283  528.34 
.7925  792.51 


4  2441  141.258     5.232  1.0567  1056.68 

5  .3051  176.572     6.540  1.3209  1320.85 

6  .3661  211.887     7.848  1.5850  1585.02 


16  387  .02832  .7646     3.7854  .00379 

32.774  .05663  1.5291     7.5709  .00757 

49.161  .08495  2.2937  11.3563  .01136 

65  549  .11327  3.0582  15.1417  .01514 

8L936  .14159  3.8228  18.9272  .01893 

98.323  .16990  4.5874  22.7126  .02271 


CONVERSION   OF   UNITS 
METRIC  TO   U.  S.  AND    U.   S.   TO    METRIC 

Weights 


77 


No, 

MS. 
to 
Gr. 

G. 

to 
Ounce 

Kg. 
to 
Lb. 

Tonnes  to 
Tons  of 
2000  Ib 

Gr. 
to 
MS. 

Ounce 
to 
G. 

Lb. 
to 
Kg. 

Tons  of 
2000  Ib. 
to  Tonnes 

1 

2 
3 

.01543 
.03086 
.04630 

.03527 
.07055 
.10582 

2.2046 
4.4092 
6.6139 

1.1023 
2.2046 
3.3069 

64.799 
129.598 
194.397 

28.350 
56.699 
85.049 

.4536 

.9072 
1.3608 

.9072 
1.8144 
2.7215 

4 
5 
6 

.06173 
.07716 
.09259 

.14110 
.17637 
.21164 

8.8185 
11.0231 
13.2277 

4.4092 
5.5116 
6.6139 

259.196 
323.995 

388.794 

113.398 
141.748 
170.097 

1.8144 
2.2680 
2.7215 

3.6287 
4.5359 
5.4431 

7 

8 
9 

.10803 
.12346 

.13889 

.24692 
.28219 
.31747 

15.4324 
17.6370 
19.8416 

7.7162 

8.8185 
9.9208 

453.592 
518.391 
583.190 

198.467 
226.796 
255.146 

3.1751 

3.6287 
4.0823 

6.3503 

7.2575 
8.1647 

Pressures 

No. 

Kg.  per 
Lin.  M. 
to 
Lb.  per 
Lin.  Ft. 

Kg.  per 
Sq,  Cm. 
to 
Lb.  per 
Sq.  In. 

KS-  per 
Sq.  M. 
to 
Lb.  per 
Sq,  Ft. 

Tonnes  per 
Sq.  M. 
to 
Tons  20001b 
per  Sq.  Ft. 

Lb.  per 
Lin.  Ft. 
to 
Kg.  per 
Lin.  M. 

Lb.  per 
Sq.  In. 
to 
KS.  per 
Sq.  Cm. 

Lb.  per  Tons  20001b 
Sq.  Ft.      per   Sq.  Ft. 
to                  to 
Kg-  per        Tonnes 
Sq.  M.     perSq.    M. 

1 

2 
3 

.6720 
1.3439 
2.0159 

14.223 
28.447 
42.670 

.2048 
.4096 
.6145 

.1024 
.2048 
.3072 

1.4882 
2.9763 
4.4645 

.07031 
.14061 
21092 

4.8824 
9.7648 
14.6472 

9.765 
19.530 
29.294 

4 
5 
6 

2.6879 
3.3598 
4.0318 

56.894 
71.117 
85.340 

.8193 
1.0241 
1.2289 

.4096 
.5120 
.6145 

5.9526 
7.4408 
8.9290 

.28123 
.35153 
.42184 

19.5296 
24.4120 
29.2944 

39.059 

48.824 
58.589 

7 

8 
9 

4.7038 
5.3757 
6.0477 

99.564 
113.787 
128.011 

1.4337 
1.6385 
1.8433 

.7169 
.8193 
.9217 

10.4171 
11.9053 
13.3934 

.49215 
.56245 
.63276 

34.1769 
39.0593 
43.9417 

68.354 
78.119 

87.883 

Miscellaneous 

G.  per 

Cu.  Cm. 
No.        to 
Lbs.  per 
Cu.  Ft. 

KS.  per 
Cu.  M. 
to 
Lbs-  per 
Cu.  Ft. 

Kg.-M. 
to 
Ft.-Lb. 

Metric 
H.  P. 
to 
U.  S. 
H.  P. 

Lbs.  per 
Cu.  Ft. 
to 
G.  per 
Cu.  Cm. 

Lbs.  per 
Cu.  Ft. 
to 
Kg.  per 
Cu.  M. 

16.018 
32.037 
48.056 

Ft.-lb. 
to 
Kg.-M. 

U.  S. 
H.  P. 

to 
Metric 
H.  P. 

1 
2 

3 

62.43 
124.86 
187.28 

.06243 
.12486 

.18728 

7.233 
14.466 
21.699 

.9863 
1.9726 
2.9589 

.01602 
.03204 
.04806 

.1383 
.2765 
.4148 

1.0139 
2.0Z77 
3.0416 

4 
5 

6 

249.71 
312.14 
374.57 

.24971 
.31214 

.37457 

28.932 
36.165 
43.398 

3.9453 
4.9316 
5.9179 

.06407 
.08009 
.09611 

64.073 
80.092 
96.110 

.5530 
.6913 
.8295 

4.0555 
5.0694 
6.0832 

7 
8 
9 

437.00 
499.43 
561.85 

.43700 
.49943 
.56185 

50.631 
57.864 
65.097 

6.9042 
7.8905 
8.8769 

.11213 
.12815 
.14416 

112.129 
128.147 
144.165 

.9678 
1.1060 
1.2443 

7.0971 
8.1110 
9.1249 

78 


CONVERSION   FACTORS 


To  reduce  N  units  of: 

To  M  units  of: 

Multiply  N  by: 

Log 

Angles 

Min. 

rad. 

.00029089 

6.46373 

Deg. 

rad. 

7T 

.017453 
.005556 

8.24187 
7.74473 

Rad. 

sec. 
min. 
deg. 

7T 

206265. 
3437.75 
57.2958 
.31831 

5.31443 
3.53627 

1.75812 
9.50285 

7T 

rad. 

3.14159 

0.49715 

Lengths 

In. 

mil. 
ft. 
yd. 
cm. 

1000. 
.083333 
.027778 
2.54 

3.00000 
8.92082 
8.44370 
0.40483 

Ft. 

in. 

yd. 
chain  (Gunter's) 
mile 
cm. 
m. 

12. 
.33333 
.015152 
.00018939 
30.48 
.3048 

1.07918 
9.52288 
8.18046 
6.27737 
1.48402 
9.48402 

Yd. 

in. 

ft. 
mile 
m. 

36. 
3. 

.00056818 
.9144 

1.55630 
0.47712 
6.75449 
9.96114 

Mile 

in. 
ft. 

yd. 

km1. 

63360. 
5280. 
1760. 
1.60935 

4.80182 
3.72263 
3.24551 
0.20665 

Link   (Gunter's) 
Chain   (Gunter's) 

in. 
ft. 

7.92 
66. 

0.89873 
1.81954 

Cm. 

in. 
ft. 

.3937 

.032808 

9.59517 

8.51598 

M. 

in. 
ft. 
yd. 

39.37 
3.2808 
1.0936 

1.59517 
0.51598 
0.03886 

Km. 

ft. 
mile 

3280.8 
.62137 

3.51598 
9.79335 

Areas                                                                                              ^ 

Cir.  mils 

sq.   mils 
sq.   mm. 

.7854 
.00050671 

9.89509 
6.70476 

Sq.  in. 

cir.   mils 
sq.   ft. 
sq.  cm. 

1273240. 
.0069444 
6.4516 

6.10491 
7.84164 
0.80967 

Sq.   ft. 

sq.   in. 
sq.  yd. 
sq.  m. 

144. 
.11111 
.0929 

2.15836 
9.04576 
8.96803 

Sq.  yd. 

sq.   ft. 
sq.  m. 

9. 
.83613 

0  95424 
9.92227 

CONVERSION 

FACTORS 

79 

To  reduce  N 

units  of:    To  M  units  of: 

Multiply  N  by; 

Log. 

Areas  —  Con't. 

Acres 

sq.  rt. 
sq.   yd. 
sq.  miles 
sq.  m. 
hectares 

43560. 
4840. 
.0015625 
4046.87 
.404687 

4.63909 
3.68485 
7.19382 
3.60712 
9.60712 

Sq.  miles 

acres 
hectares 
sq.   km. 

640. 
259. 
2.59 

2.80618 
2.41330 
0.41330 

Sq.   mm. 

cir,  mils 
sq.  in. 

1973.52 
.00155 

3.29524 
7.19033 

Sq.    cm. 

sq.  in. 
sq.   ft. 

.155 
.0010764 

9.19033 
7.03197 

Sq.  m'. 

sq.   ft. 
sq.   yd. 
acres 

10.7639 
1.196 
.0002471 

1.03197 
0.07773 
6.39288 

Hectares 

acres 
sq.   miles 

2.471 
.003861 

0.39288 
7.58670 

Sq.   km. 

sq.  miles 

.3861 

9.58670 

Volumes 

Cu.  in. 

cu.  ft. 
cu.  cm. 
gal. 
liters 

.0005787 
16.387 
.004329 
.016387 

6.76246 
1.21450 
7.63639 
8.21450 

Cu.  ft. 

cu.  in. 
cu.  yd. 
cu.  cm. 
cu.  m. 
gal. 
liters 

1728. 
.03778 
28317. 
.028317 
7.4805 
28.317 

3.23754 
8.56864 
4.45205 
8.45205 
0.87393 
1.45205 

Cu.  yd. 

cu.   in. 
cu.  ft. 
acre-ft. 
cu.  m. 

46656. 

27. 
.0006198 
.76456 

4.66891 
1.43136 

6.79228 
9.88341 

Gal. 

cu.   in. 
cu.  ft. 
cu.   cm1, 
liters 
cu.  m. 

231. 
.13368 
3785.4 
3.7854 
.0037854 

43560. 
1613.33 
325851. 
1233.49 

2.36361 
9.12607 
3.57812 
0.57812 
7.57812 

Acre-ft. 

cu.  ft. 
cu.  yd. 
gal. 
cu.  m. 

4.63909 
3.20772 
5  .  51302 
3.09114 

Cu.  cm. 

cu.  in. 
cu.  ft. 
gal. 

.061023 
.000035314 
.00026417 

8.78550 
5.54795 
6.42188 

Cu.   m. 

cu.  ft. 
cu.  yd. 
gal. 

35.3145 
1.3079 
264.17 

1.54795 
0.11659 
2.42188 

SI) 


CONVERSION  FACTORS 


To  reduce  N  units  of  :    To  M  units  of  : 

Multiply  N  by: 

Log. 

Volumes  —  Con 

't. 

Liters 

cu.  in. 
cu.  ft. 
gal. 

61.023 
.0.35314 

.2*6417 

1.78550 
8.51795 
9.42188 

Water   Discharges 

Cu.  ft.  per  sec. 

gal.  per  min. 
gal.  per  hr. 
gal.  per  day 
acre-ft.  per  day 
miner's  in. 

448.83 
26930. 
646317. 
1.9835 
40. 

2.65208 
4.43024 
5.81045 
0.29743 
1.60206 

Cu.   ft.  per  min. 
Cu.  ft.  per  hr. 

gal.  per  hr. 
gal.  per  day 

448.83 
179.53 

2.65208 
2.25414 

Cu.  ft.  per  day 

gal.  per  min. 
gal.  per  hr. 

.005195 
.31169 

7.71559 
9.49372 

Gal.  per  sec. 

cu.  ft.  per  hr. 
acre-ft.  per  day 

481.25 
.26515 

2.68237 
9.42349 

Gal.  per  min. 

cu.  ft.  per  sec. 
cu.  ft.  per  hr. 
acre-ft.  per  day 
miner's  in. 

.002228 
8.0208 
.004419 

.08912 

7.34792 

0.90422 
7.64534 
8.94998 

Miner's  in. 

cu.  ft.  per  sec. 
cu.  ft.  per  min. 
gal.  per  min. 

.025 
1.5 
11.22 

8.39794 
0.17609 
1.05002 

1000000  gal.  per  day 

cu.  ft.  per  sec. 
gal.  per  min. 
miner's  in. 

1.547 
694.4 
61.88 

0.18955 
2.84164 
1.79155 

Acre-ft.  per  day 

cu.  ft.  per  sec. 
gal.  per  min. 
miner's  in. 

.5042 
226.3 
20.17 

9.70257 
2.35466 
1.30463 

Weights 

Gr. 

g. 

.064799 

8.81157 

Ounce 

£• 

g. 

437.5 
.0625 
28.3496 

2.64098 
8.79588 
1.45255 

Lb. 

ounce 

Is. 

16. 
453.593 
.453593 

1.20412 
2.65667 
9.65667 

Ton   (2240  Ib.) 
Ton   (2000  Ib.) 

metric  ton 
metric  ton 

1.01605 
.90719 

0.00691 
9.95770 

G. 

gr. 
ounce 
Ib. 

15.43235 
.035274 
.0022046 

1.18843 
8.54745 
7.34333 

Kg. 
Metric  ton 
Metric  ton 

Ib. 
ton   (2000  Ib.) 
ton   (2240  Ib.) 

2.2046 
1.1023 
.98421 

0.34333 
0.04230 
9.99309 

CONVERSION   FACTORS 

81 

To  reduce  N  units  of:     To  M  units  of: 

Multiply  N  by 

:        Log. 

Weights  per  Unit 

Length 

Lb.  per  in. 
Lb.  per  ft. 
G.  per  cm. 
Kg.   per  m. 

g.  per  cm. 
kg.  per  m. 
Ib.  per  in. 
Ib.  per  ft. 

17.858 
1.48816 
.056 
.67197 

1.25183 
0.17265 
8.74817 
9.82735 

Densities 

Lb.  per  cu.  in. 
Lb.  per  cu.  ft. 
G.  per  cu.  cm. 
Kg.  per  cu.  m. 

g.  per  cu.  cm. 
kg.  per  cu.  m. 
Ib.  per  cu.  in. 
Ib.  per  cu.  ft. 

27.6797 
16.018 
.036128 
.062428 

1.44216 
1.20462 
8.55784 
8.79538 

Water  —  Volumes  to 

Welgfits 

Cu.  in. 

ounces 
Ib. 
g. 

.578 
.03613 
16.387 

9.76196 
8.55784 
1.21450 

Cu.  ft. 

Ib. 
kg. 

62.428 
28.317 

1.79538 
1.45205 

Cu.  yd. 

Ib. 

1686. 

3.22675 

Cu.   cm. 

ounces 
Ib. 
g. 

.03527 
.0022046 
1. 

8.54745 
7.34333 

0.00000 

Cu.  m. 

Ib. 

2205. 

3.34333 

Gal. 

Ib. 
kg. 

8.345 
3.785 

0.92145 
0.57812 

Liters 

Ib. 

2.205 

0.34333 

Water—  Weights  to 

Volumes 

• 

Ounces 

cu.  in. 
cu.  cm. 

1.73 
28.35 

0.23804 
1.45255 

Lb. 

cu.  in. 

cu.  ft. 
cu.  cm. 
gal. 
liters 

27.68 
.016018 
453.59 
.1198 
.45359 

1.44216 
8.20462 
2.65667 
9.07855 
9.65667 

G. 

cu.   in. 

.06102 

8.78550 

Kg. 

cu.  ft. 
gal. 

.0353 
.2642 

8.54795 
9.42188 

Forces 

Lb.-wt. 
G.-wt. 

dynes 
dynes 

444793. 
980.6 

5.64816 
2.9914& 

Dynes 

Ib.-wt. 
g.-  wt. 

2.248xlO-6 
.00102 

4.35184 
7.00851 

CONVERSION   FACTORS 


To  reduce  N  units  of: 

To  M  units  of: 

Multiply  N  by: 

Loar. 

Pressures 

Lb.  per  sq.  in. 

cm.  of  mercury 
in.  of  mercury 
ft.  of  water 
kg.  per  sq.  cm. 
atmospheres 

5.1712 
2.0359 
2.3067 
.070307 
.068041 

0.71359 

0.30875 
0.36298 
8.84700 
8.83277 

Lb.  per  sq.  ft. 

ft.  of  water 
kg.  per  sq.  m. 

.016018 
4.8824 

8.20462 
0.68863 

Ft.  of  water 

Ib.   per  sq.  in. 
Ib.  per  sq.  ft. 
kg.  per  sq.  cm. 
atmospheres 
in.  of  mercury 
cm.  of  mercury 

.43353 

62.43 
.03048 
.  0295 
.88261 
2.2412 

9.63702 
1.79538 
8.48402 
B  46979 
9.94577 
0.35048 

In.  of  mercury 

Ib.  per  sq.  in. 
ft.  of  water 
kg.  per  sq.  cm. 
atmospheres 

.49119 
1.133 
.034534 
.03342 

9.69125 
O.C5423 
8.53S25 
8.52402 

<?m.  of  mercury 

Ib.  per  sq.  in. 
ft.  of  water 
kg.  per  sq.  cm. 
atmospheres 
dynes  per  sq.  cm. 

.1934 
.4462 
.0136 
.01316 
13336. 

9.28641 
9.64952 
8.13354 
8.11919 
4.12503 

Kg",  per  sq.  cm. 
Kg.  per  sq.  m. 

Ib.  per  sq.  in. 
in.  of  mercury 
cm.  of  mercury 
ft.  of  water 
atmospheres 

14.223 
28.957 
73.53 

32.808 
.96778 

1.15300 
1.46175 
1.86646 
1.51598 
9.98578 

Ib.  per  sq.  ft. 

.20482 

9.31137 

Atmospheres 

• 

Ib.  per  sq.  in. 
in.  of  mercury 
cm',   of  mercury 
ft.  of  water 
kg.  per  sq.  cm. 

14.697 
29.921 
76. 
33.9 
1.0333 

1.16723 
1.47598 
1.88081 
1.53021 
0.01422 

Energy,  Work 

,   Heat 

In.-lb. 

kg.  -cm. 

1.152 

0.06150 

Ft.-lb. 

g.-cm. 
kg.-m. 
joules 
watt-hr. 
calories 
B.  T.  U. 

13825. 
.13825 
1.3563 
.0003767 
.3239 
.001285 

4.14068 
9.14068 
0.13235 
6.57605 
9.51041 
7.10902 

Kg.  -cm. 

in.-lb. 

.868 

9.93850 

Kg.-m. 

ft.-lb. 
H.  P.-sec. 
joules 
watt-hr. 
calories 
B.  T.  U. 

7.233 
.01315 
9.81 
.002725 
2.3428 
.009297 

0.85932 
8.11896 
0.99167 
7.43537 
0.36973 
7.96834 

CONVERSION   FACTORS 


To  reduca  N  unit  s  of.    To  M  units  of: 

Multiply  N  by: 

Log. 

Energy,   Work,    Heat—  Con't. 

Joule,  or  watt-sec. 

g.-cm. 
ft.-lb. 
ergs 
calorie 

10198. 
.7373 
107 
.2389 

4.00852. 
9.86765 
7.00000 
9.378^3. 

Watt-hr. 

ft.-lb. 
kg.-m. 
calories 
B.  T.  IT. 

2654.3 
366.97 
859.74 
3.4117 

3.42395- 
2.56463 
2.934<>  7 
0.53297 

H.  P.  -sec. 

kg.-m. 
calories 

76.04 
178.15 

1.88104 

2.25078 

H.  P.-hr. 

ft.-lb. 
B.  T.  U. 

1980000. 
2545. 

6.2966T 
3.4056D 

Calorie 

ft.-lb. 
kg.-m. 
joule 
B.  T.  U. 

3.087 
.42685 
4.1857 
.003968 

0.48959 
9.6302T 
0.62182 
7.59861 

B.  T.  U. 

ft.-lb. 
kg.-m. 
watt-hr. 
H.  P.-hr. 
calorie 

778.1 
107.56 
.293 
.0003929 
252. 

2.89098 
2.0316S 
9.46703 
6.59431 
2.40140' 

Power 

Ft.-lb.  per  sec. 

watts 
H.  P. 
kw. 
metric  H.  P. 

1.356 
.001818 
.001356 
.001843 

0.13235 
7.25964 
7.13235 
7.26562 

Kg.-m.  per  sec. 

watts 
H.P. 
metric  H.P. 

9.81 
.0315 
.01333 

0.99167 
8.11896 
8.12494 

Watts 

ft.-lb.  per  sec. 
kg.-m.  per  sec. 
H.P. 
metric  H.P. 

.7373 
.10194 
.00134 
.00136 

9.86765 
9.00833 
7.12729 
7.13327 

Kw. 

ft.-lb.  per  sec. 
ft.-lb.  per  min. 
kg.-m.  per  min. 
H.P. 
metric  H.P. 
B.  T.  U.  per  min. 
B.  T.  U.  per  hr. 

737.3 
44239. 
6116.4 
1.341 
1.359 
56.86 
3411.7 

2.86765 
4.64580 
3.78650 
0.12729 
0.13327 
1.75481 
3.53297 

H.    P. 

ft.-lb.  per  sec. 
ft.-lb.  per  min. 
watts 
kw. 
metric  H.P. 
B.  T.  U.  per  min. 
B.  T.  U.  per  hr. 

550. 
33000. 
746. 
.746 
1.0139 
42.416 
2545. 

2.74036 
4.51851 
2.87271 
9.87271 
0.00598 
1.62753 
3.4056&' 

M 


CONVERSION   FACTORS     ' 


To  reduce  N  units  of:    To  M  units  of: 

Multiply  N  by: 

^Log. 

Power  —  Con't. 

Metric  H.  P.                     ft.lb.  per  sec. 
kg.-m.  per  sec. 
watts 
kw. 
H.P. 

542.47 
75. 
735.75 
.73575 

.9863 

2.73438 
1.87506 
2.86673 
9.86673 
9.99402 

B.  T.  U.  per.  min.          kw. 
H.P. 

.01759 
.02358 

8.24519 
8.37247 

B.  T.  U.  per  hr.               kw. 
H.P. 

.000293 
.000393 

6.46703 
6.59431 

Angular  Velocities 

Rad.  per  sec.                    rev.  per  sec. 
rev.  per  min. 

.15916 
9.5496 

9.20182 
0.97997 

Rad.  per  min.                   rev.  per  sec. 

.002653 

7.42367 

Rev.  per  min.                   rad.  per  sec. 
deg.   per  sec. 

.10472 
6. 

9.02003 
0.77815 

Deg.  per  sec.                    rev.  per  min. 

.16667 

9.22185 

Linear   Velocities 

Ft;-  per  sec.                        m.  per  min. 
Ft.   per  min.                       miles  per  hr. 

18.288 
.01136 

1.26217 
8.05552 

Miles  per  hr.                      ft.  per  min. 
m.  per  min. 

88. 

26.82 

1.94448 
1.42850 

M.  per  min.                       ft.  per  sec. 
miles  per  hr. 

.0547 

.03728 

8.73783 
8.57150 

Temperatures 

Deg.  C.                                 (deg.  Fah.—  32) 
(Deg-.   Fah.—  32)               deg.  C. 

1.8 
.55556 

0.25527 
9.74473 

Electrostatic  to  Electromagnetic 

Potential                            volts 
Capacitv                             microfarads 
Quantity                              coulombs 
Current                               amperes 
Resistance                         ohms 

300. 
l.llllxlO-fi 
3.333xlO-io 
3.333xlO~10 
9.X1011 

2.47712 
4.04575 
0.52287 
0.52287 
11.95424 

WIRE  AND  SHEET  METAL  GAUGES 
In  Decimals  of  an  Inch 


& 

c 

1 

CS 

J5  . 

J 

T3  "ft  — 

"c  rt  w 

8   d 

ii! 

6 

6 

El 

te 

S«3 

-M  t)  '& 

2  he  °o 

c 

oj 

c 

.~  i" 

£/2  Q)  C 

,0   .S 

0 

CJ 

•S  'fi 

o 

i 

i 

£ 

<  w 

w  *  c 

rill 

III 

1 

c  £j 
<J  CQ 

l| 

pq  £_! 

7-0 

.5 

.500 

6-0 

.46875 

.4600 

.464 

5-0 

.4375 

.4300 

.45 

.432 

•  4-0 

.454 

.46000 

.40625 

.3938 

.40 

.400 

3-0 

.425 

.40964 

.375 

.3625 

.36 

.0315 

.372 

2-0 

.380 

.36480 

.34375 

.3310 

.33 

.0447 

.348 

0 

.340 

.32486 

.3125 

.3065 

.305 

.0578 

.324 

1 

.300 

.28930 

.28125 

.2830 

.285 

.0710 

.300 

2 

.284 

.25763 

.26562 

.2625 

.265 

.0842 

.276 

3 

.259 

•  .22942 

.25 

.2437 

.245 

.0973 

.252 

4 

.238 

.20431 

.23437 

.2253 

.225 

.1105 

.232 

5 

.220 

.18194 

.21875 

.2070 

.205 

.1236 

212 

6 

.203 

.16202 

.20312 

.1920 

.190 

.1368 

!l92 

7 

.180 

.14428 

.1875 

.1770 

.175 

.1500 

.176 

8 

.165 

.12849 

.17187 

.1620 

.160 

.1631 

.160 

9 

.148 

.11442 

.15625 

.1483 

.145 

.1763 

.144 

10 

.134 

.10190 

.14062 

.1350 

.130 

.1894 

.128 

11 

.120 

.09074 

.125 

.1205 

.1175 

.2026 

.116 

12 

.109 

.08081 

.10937 

.1055 

.1050 

.2158 

.104 

13 

.095 

.07196 

.09375 

.0915 

.0925 

.2289 

.092 

14 

.083 

.06408 

.07812 

.0800 

.0806 

.2421 

.080 

15 

.072 

.05707 

.07031 

.0720 

.0700 

.2552 

.072 

16 

.065 

.05082 

.0625 

.0625 

.0610 

.2684 

.064 

17 

.058 

.04526 

.05625 

.0540 

.0525 

.2816 

.056 

18 

.049 

.04030 

.05 

.0475 

.0450 

.2947 

.048 

19 

.042 

.03589 

.04375 

.0410 

.0400 

.3079 

.040 

20 

.035 

.03196 

.0375 

.0348 

.0350 

.3210 

.036 

21 

.032 

.02846 

.03437 

.0317 

.0310 

.3342 

.032 

22 

.028 

.02535 

.03125 

.0286 

.0280 

.3474 

.028 

23 

.025 

.02257 

.02812 

.0258 

.0250 

.3605 

.024 

24 

.022 

.02010 

.025 

.0230 

.0225 

.3737 

.022 

25 

.020 

.01790 

.02187 

.0204 

.0200 

.3868 

.020 

26 

.018 

.01594 

.01875 

.0181 

.0180 

.4000 

.018 

27 

.016 

.01419 

.01719 

.0173 

.0170 

.4132 

.0164 

28 

.014 

.01264 

.01562 

.0162 

.0160 

.4263 

.0148 

29 

.013 

.01126 

.01406 

.0150 

.0150 

.4395 

.0136 

30 

.012 

.01002 

.0125 

.0140 

.0140 

.4526 

.0124 

31 

.010 

.00893 

.01094 

.0132 

.0130 

.4658 

.0116 

32 

.009 

.00795 

.01016 

.0128 

.0120 

.4790 

.0108 

33 

.008 

.00708 

.00937 

.0118 

.0110 

.4921 

.01-jO 

34 

.007 

.00630 

.00859 

.0104 

.0100 

.5053 

.0092 

35 

.005 

.00561 

.00781 

.0095 

.0095 

.5184 

.0084 

36 

.004 

.00500 

.00703 

.0090 

.0090 

.5316 

.0076 

37 

.00445 

.00664 

.0085 

.0085 

.5448 

.0068 

38 

.00396 

.00625 

.0080 

.0080 

.5579 

.0060 

39 

.00353 

.0075 

.0075 

.5711 

.0052 

40 

.00314 

.0070 

.0070 

.5842 

.0048 

POWERS  AND  MULTIPLES  OF  TT  e    AND 


Powers  and  Multiples  of  TT  e  and  g. 


Quantity 

Value           Log. 

Quantity                Value 

Log. 

7T 

3.14159  0.49715 

e                  2.71828 

0.43429 

27T 

6.28319  0.79818 

e2                7.38906 

0.86859 

3   7T 

9.42478  0.97427 

•      l~e             0.36788 

9.56571 

4   7T 

12.56637  1.09921 

Ve               1.64872 

0.21715 

4  7T   --3 

4.18879  0.62209 

*981. 

2.99167 

9               32  17 

1.50745 

7T   -r-   2 

1.57080  0.19612 

*    0.0005097 

6.70730 

1TH-3 

1.04720  0.02003 

1  ~*~  2#            0.01554 

8.19152 

7T  -f-   4 

0.78540  9.89509 

*  31.3209 

1.49583 

37T-T-2 

4.71239  0.67324 

^9               5.6719 

0  .  75373 

1   -i-    7T 

0.31831  9.50285 

*  44.2945 

1.64635 

7T2 

9.86960  0.99430 

X^0              8.0212 

0.90424 

47T2 

39.47842  1.59636 

_*  62.6418 

1.79686 

7T2  —  4 

2.46740     .39224 

2  ^9            11.3438 

1.05476 

7T:{ 

31.00628  1.49145 

_*    0.02258 

8  35365 

97.40909  1.98860 

1  -s-  V  2g          0.1247 

9.09576 

/ 

1.77245  0.24857 

_«  98.3976 

1.99298 

V 

*  v'  9             17.8187 

1  .  25088 

7TX/7T 

5.56833  0.74572 

*139.1536 

2.14350 

7TX/2 

4.44288  0.64766 

""  N/  %9          25  .  1994 

1.40139 

2x/7T— 

3.54491  0.549CO 

_«    0.1003 

9.00132 

V'Tir 

2.50663  0.39909 

""  -s-  v7  g          0  .  5539 

9.74342 

T  4-  >/~2~~ 

2.22144  0.34663 

*    0.07093 

8.85080 

\/7r^2 

1.25331  0.09806 

ir~\/2g           0.39166 

9.59291 

%/^FiT 

0.797889.90194 

fy-$-^-Tf*q       0.4789 

9.68027 

S/-— 

1.46459  0.16572 

A9  ^ 

' 

4.60115  0.66287 

b^:^            0.001766 

7  .  24704 

7T  ^      7T 

tf  2^7T~ 

1  84526  0.26606 

CO    K 

r»«  • 

2.14503  0.33143 

bZ-°               1.9428 
0 

0.28843 

NOTE:—  Negative  characteristics  of  logarithms  have  been 

increased  by 

10  and  are  shown 

in  black  type. 

*    Metric  units 

HYPERBOLIC   FUNCTIONS 


S7 


HYPERBOLIC  FUNCTIONS 

Formulae 
*sinhw=    e?—-*~.          cosh  w=    e '--' 


2 


tanh  u 


sinh  u   _  e"  — e  —u 

cosh  u       e  u-\-e  —u 

sinh  ( — M)  =  —  sinh  u 
tanh  ( —  w)  =  —  tanh  u 
cosh—1  u  =ln  (w-j-N/w2 — 1) 


cosh2  w— sinh2  u  =  l 
cosh  ( — w)  =  -f-  cosh  u 
sinh-1  u  =  In  (w-f -v/w2-f  1 

tanh— 1u=  In 

Since  cosh  u  -f-  sinh  u  =  eu  and  cosh  u — sinh  u=e~ ",  the  value 
of  e"  and  e~u  are  readily  obtained  from  the  tables  for  sinh  u  and 
cosh  u. 

*  Read  hyperbolic  sine  of  u 


Tanh  u 

tt=0 

to  u= 

2.29 

u 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

0 

.1 

.2 
.3 

0000 
.0997 
.1974 
.2913 

.0100 
.1096 
.2070 
.3004 

.0200 
.1194 
.2165 
.3095 

.0300 
.1293 
.2260 
.3185 

.0400 
.1391 
.2355 
.3275 

.0500 
.1489 
.2449 
.3364 

.0599 
.1587 
.2543 
.3452 

.0699 
.1684 
.2636 
.3540 

.0798 
.1781 
.2729 
.3627 

.0898 
.1878 
.2821 
.3714 

.4 
.5 
.6 

.3800 
.4621 
.5370 

.3885 
.4700 
.5441 

.3969 

.4777 
.5511 

.4053 
.4854 
.5581 

.4137 
.4930 
.5649 

.4219 
.5005 
.5717 

.4301 

.5080 
.5784 

.4382 
.5154 
.5850 

.4462 
.5227 
.5915 

.4542 
.5299 
.5980 

.  7 
.8 
.9 

.6044 
.6640 
.7163 

.6107 
.6696 
.7211 

.6169 
.6751 
.7259 

.6231 
.6805 
.7306 

.6291 
.6858 
.7352 

.6352 
.6911 
.7398 

.6411 
.6963 
.7443 

.6469 
.7014 

.7487 

.6527 
.7064 
.7531 

.6584 
.7114 
.7574 

1.0 
.1 
2 

'.3 

.7616 
.8005 
.8337 
.8617 

.7658 
.8041 
.8367 
.8643 

.7699 

.8076 
.8397 
.8668 

.7739 
.8110 
.8426 
.8693 

.7779 
.8144 
.8455 
.8717 

.7818 
.8178 
.8483 
.8741 

.7857 
.8210 
.8511 

.8764 

.7895 
.8243 
.8538 
.8787 

.7932 

.8275 
.8565 
.8810 

.7969 
.8306 
.8591 
.8832 

.4 
.5 
.6 

.8854 
.9052 
.9217 

.8875 
.9069 
.9232 

.8896 
.9087 
.9246 

.8917 
.9104 
.9261 

.8937 
.9121 
.9275 

.8957 
.9138 
.9289 

.8977 
.9154 
.9302 

.8996 
.9170 
.9316 

.9015 
.9186 
.9329 

.9033 
.9202 
.9342. 

.7 
.8 
.9 

.9354 
.9468 
.9562 

.9367 
.9478 
.9571 

.9379 
.9488 
.9579 

.9391 
.9498 
.9587 

.9402 
.9508 
.9595 

.9414 

.9518 
.9603 

.9425 
.9527 
.9611 

.9436 
.9536 
.9619 

.9447 
.9545 
.9626 

.9458 
.9554 
.9633 

2.0 

.1 

2 

.9640 
.9705 
.9757 

.9647 
.9710 
.9762 

.9654 
.9716 
.9767 

.9661 
.9722 
.9771 

.9668 
.9727 
.9776 

.9674 
.9732 

.9780 

.9680 
.9738 

.9785 

.9687 
.9743 
.9789 

.9693 
.9748 
.9793 

.9699 
.9753 
.9797 

HYPERBOLIC  FUNCTIONS 


Sinh  u 


=  0  to  t<  =  4. 


0  .0000  .0100  .0200  .0300  .0400 

.1  .1002  .1102  .1203  .1304  .1405 

.2  .2013  .2115  .2218  .2320  .2423 

.3  .3045  .3150  .3255  .3360  .3466 

.4  .4108  .4216  .4325  .4434  .4543 

.5  .5211  .5324  .5438  .5552  .5666 

.6  .6367  .6485  .6605  .6725  .6846 

.7  .7586  .7712  .7838  .7966  .8094 

.8  .8881  .9015  .9150  .9286  .9423 

.9  1.0265  1.0409  1.0554  1.0700  1.0847 

1.0  1.1752  1.1907  1.2063  1.2220  1.2379 

.1  1.3356  1.3524  1.3693  1.3863  1.4035 

.2  1.5095  1.5276  1.5460  1.5645  1.5831 

.3  1.6984  1.7182  1.7381  1.7583  1.7786 

.4  1.9043  1.9259  1.9477  1.9697  1.9919 

.5  2.1293  2.1529  2.1768  2.2008  2.2251 

.6  2.3756  2.4015  2.4276  2.4540  2.4806 

.7  2.6456  2.6740  2.7027  2.7317  2.7609 

.8  2.9422  2.9734  3.0049  3.0367  3.0689 

.9  3.2682  3.3025  3.3372  3.3722  3.4075 

2.0  3.6269  3.6647  3.7028  3.7414  3.7803 

.1  4.0219  4.0635  4.1056  4.1480  4.1909 

.2  4.4571  4.5030  4.5494  4.5962  4.6434 

.3  4.9370  4.9876  5.0387  5.0903  5.1425 

.4  5.4662  5.5221  5.5785  5.6354  5.6929 

.5  6.0502  6.1118  6.1741  6.2369  6.3004 

.6  6.6947  6.7628  6.8315  6.9009  6.9709 

.7  7.4063  7.4814  7.5572  7.6338  7.7112 

.8  8.1919  8.2749  8.3586  8.4432  8.5287 

.9  9.0596  9.1512  9.2437  9.3371  9.4315 

3.0  10.018  10.119  10.221  10.324  10.429 

.1  11.076  11.188  11.301  11.415  11.530 

.2  12.246  12.369  12.494  12.620  12.747 

.3  13.538  13.674  13.812  13.951  14.092 

.4  14.965  15.116  15.268  15.422  15.577 

.5  16.543  16.709  16.877  17.047  17.219 

.6  18.285  18.470  18.655  18.843  19.033 

.7  20.211  20.415  20.620  20.828  21.037 

.8  22.339  22.564  22.791  23.020  23.252 

.9  24.691  24.939  25.190  25.444  25.700 

4.0  27.290  27.564  27.842  28.122  28.404 

.1  30.162  30.465  30.772  31.081  31.393 

.2  33.336  33.671  34.009  34.351  34.697 

.3  36.843  37.214  37.588  37.966  38.347 

.4  40.719  41.129  41.542  41.960  42.382 

.5  45.003  45.455  45.912  46.374  46.840 

.6  49.737  50.237  50.742  51.252  51.767 

7  54.969  55.522  56.080  56.643  57.213 

.8  60.751  61.362  61.979  62.601  63.231 

.9  67.141  67.816  68.496  69.186  69.882 


.0500  .0600  .0701  .0801  .0901 

.1506  .1607  .1708  .1810  .1911 

.2526  .2629  .2733  .2837  .2941 

.3572  .3678  .3785  .3892  .4000 

.4653  .4764  .4875  .4986  .5098 

.5782  .5897  .6014  .6131  .6248 

.6967  .7090  .7213  .7336  .7461 

.8223  .8353  .8484  .8615  .8748 

.9561  .9700  .9840  .9981  1.0122 

1.0995  1.1144  1.1294  1.1446  1.1598 

1.2539  1.2700  1.2862  1.3025  1.3190 

1.4208  1.4382  1.4558  1.4735  1.4914 

1.6019  1.6209  1.6400  1.6593  1.6788 

1.7991  1.8198  1.8406  1.8617  1.8829 

2.0143  2.0369  2.0597  2.0827  2.1059 

2.2496  2.2743  2.2993  2.3245  2.3499 

2.5075  2.5346  2.5620  2.589'>  2.6175 

2.7904  2.8202  2.8503  2.8806  2.9112 

3.1013  3.1340  3.1671  3.2005  3.2341 

3.4432  3.4792  3.5156  3.5523  3.5894 

3.8196  3.8593  3.8993  3.9398  3.9806 

4.2342  4.2779  4.3221  4.3666  4.4117 

4.6912  4.7394  4.7880  4.8372  4.8868 

5.1951  5.2483  5.3020  5.3562  5.4109 

5.7510  5.8097  5.8689  5.9288  5.9892 

6.3645  6.4293  6.4946  6.5607  6.6274 

7.0417  7.1132  7.1854  7.2583  7.3319 

7.7894  7.8683  7.9480  8.0285  8.1098 

8.6150  8.7021  8.7902  8.8791  8.9689 

9.5268  9.6231  9.7203  9.8185  9.9177 

10.534  10.640  10.748  10.856  10.966 

11.647  11.764  11.883  12.003  12.124 

12.876  13.006  13.137  13.269  13.403 

14.234  14.377  14.522  14.668  14.816 

15.734  15.893  16.053  16.214  16.378 

17.392  17.567  17.744  17.923  18.103 

19.224  19.418  19.613  19.811  20.010 

21.249  21.463  21.679  21.897  22.117 

23.486  23.722  23.961  24.202  24.445 

25.958  26.219  26.483  26.749  27.018 

28.690  28.979  29.270  29.564  29.862 

31.709  32.028  32.350  32.675  33.004 

35.046  35.398  35.754  36.113  36.476 

38.733  39.122  39.515  39.913  40.314 

42.808  43.238  43.673  44.112  44.555 

47.311  47.787  48.267  48.752  49.242 

52.288  52.813  53.344  53.880  54.422 

57.788  58.369  58.955  59.548  60.147 

63.866  64.508  65.157  65.812  66.473 

70.584  71.293  72.010  72.734  73.465 


HYPERBOLIC  FUNCTIONS 
Cosh  u  u  =0  to  t<=4.99 

1234  567 


89 


0  1.0000  1.0001  1.0002  1.0005  1.0008 

.1  1.0050  1.0061  1.0072  1.0085  1.0098 

.2  1.0201  1.0221  1.0243  1.0266  1.0289 

.3  1.0453  1.0484  1.0516  1.0549  1.0584 

.4  1.0811  1.0852  1.0895  1.0939  1.0984 

.5  1.1276  1.1329  1.1383  1.1438  1.1494 

.6  1.1855  1.1919  1.1984  1.2051  1.2119 

.7  1.2552  1.2628  1.2706  1.2785  1.2865 

.8  1.3374  1.3464  1.3555  1.3647  1.3740 

.9  1.4331  1.4434  1.4539  1.4645  1.4753 

1.0  1.5431  1.5549  1.5669  1.5790  1.5913 

.1  1.6685  1.6820  1.6956  1.7093  1.7233 

.2  1.8107  1.8258  1.8412  1.8568  1.8725 

.3  1.9709  1.9880  2.0053  2.0228  2.0404 

.4  2.1509  2.1700  2.1894  2.2090  2.2288 

.5  2.3524  2.3738  2.3955  2.4174  2.4395 

.6  2.5775  2.6013  2.6255  2.6499  2.6746 

.7  2.8283  2.8549  2.8818  2.9090  2.9364 

.8  3.1075  3.1371  3.1669  3.1972  3.2277 

.9  3.4177  3.4506  3.4838  3.5173  3.5512 

2.0  3.7622  3.7987  3.8355  3.8727  3.9103 

.1  4.1443  4.1847  4.2256  4.2668  4.3085 

.2  4.5679  4.6127  4.6580  4.7037  4.7499 

.3  5.0372  5.0868  5.1370  5.1876  5.2388 

.4  5.5569  5.6119  5.6674  5.7235  5.7801 

.5  6.1323  6.1931  6.2545  6.3166  6.3793 

.6  6.7690  6.8363  6.9043  6.9729  7.0423 

.7  7.4735  7.5479  7.6231  7.6990  7.7758 

.8  8.2527  8.3351  8.4182  8.5022  8.5871 

.9  9.1146  S.2056  9.2976  9.3905  9.4844 

3.0  10.068  10.168  10.270  10.373  10.476 

.1  11.121  11.233  11.345  11.459  11.574 

.2  12.287  12.410  12.534  12.660  12.786 

.3  13.575  13.711  13.848  13.987  14.127 

.4  14.999  15.149  15.301  15.455  15.610 

.5  16.573  16.739  16.907  17.077  17.248 

.6  18.313  18.497  18.682  18.870  19.059 

.7  20.236  20.439  20.644  20.852  21.061 

.8  22.362  22.586  22.813  23.042  23.273 

.9  24.711  24.959  25.210  25.463  25.719 

4.0  27.308  27.582  27.860  28.139  28.422 

.1  30.178  30.482  30.788  31.097  31.409 

.2  33.351  33.686  34.024  34.366  34.711 

.3  36.857  37.227  37.601  37.979  38.360 

.4  40.732  41.141  41.554  41.972  42.393 

.5  45.014  45.466  45.923  46.385  46.851 

.6  49.747  50.247  50.752  51.262  51.777 

.7  54.978  55.531  56.089  56.652  57.221 

.8  60.759  61.370  61.987  62.609  63.239 

.9  67.149  67.823  68.505  69.193  69.889 


1.0013  1.0018  1.0025  1.0032  1.0041 

1.0113  1.0128  1.0145  1.0162  1.0181 

1.0314  1.0340  1.0367  1.0395  1.0423 

1.0619  1.0655  1.0692  1.0731  1.0770 

1.1030  1.1077  1.1125  1.1174  1.1225 
1.1551  1.1609  1.1669  1.1730  1.1792 
1.2188  1.2258  1.2330  1.2402  1.2476 

1.2947  1.3030  1.3114  1.3199  1.3286 
1.3835  1.3932  1.4029  1.4128  1.4229 
1.4862  1.4973  1.5085  1.5199  1.5314 

1.6038  1.6164  1.6292  1.6421  1.6552 

1.7374  1.7517  1.7662  1.7808  1.7956 

1.8884  1.9045  1.9208  1.9373  1.9540 

2.0583  2.0764  2.0947  2.1132  2.1320 

2.2488  2.2691  2.2896  2.3103  2.3312 
2.4619  2.4845  2.5073  2.5305  2.5538 
2.6995  2.7247  2.7502  2.7760  2.8020 

2.9642  2.9922  3.0206  3.0492  3.0782 
3.2585  3.2897  3.3212  3.3530  3.3852 
3.5855  3.6201  3.6551  3.6904  3.7261 

3.9483  3.9867  4.0255  4.0647  4.1043 

4.3507  4.3932  4.4362  4.4797  4.5236 

4.7966  4.8437  4.8914  4.9395  4.9881 

5.2905  5.3427  5.3954  5.4487  5.5026 

5.8373  5.8951  5.9535  6.0125  6.0721 
6.4426  6.5066  6.5712  6.6365  6.7024 
7.1123  7.1831  7.2546  7.3268  7.3998 

7.8533  7.9136  8.0106  8.0905  8.1712 
8.6728  8.7594  8.8469  8.9352  9.0244 
9.5791  9.6749  9.7716  9.8693  9.9680 

10.581  10.687  10.794  10.902  11.011 

11.689  11.806  11.925  12.044  12.165 

12.915  13.044  13.175  13.307  13.440 

14.269  14.412  14.556  14.702  14.850 

15.766  15.924  16.084  16.245  16.408 
17.421  17.596  17.772  17.951  18.131 
19.250  19.444  19.639  19.836  20.035 

21.272  21.486  21.702  21.919  22.139 
23.507  23.743  23.982  24.222  24.466 
25.977  26.238  26.502  26.768  27.037 

28.707  28.996  29.287  29.581  29.878 

31.725  32.044  32.365  32.691  33.019 

35.060  35.412  35.768  36.127  36.490 

38.746  39.135  39.528  39.925  40.X26 

42.819  43.250  43.684  44.123  44.566 
47.321  47.797  48.277  48.762  49.252 
52.297  52.823  53.354  53.890  54.431 

57.796  58.377  58.964  59.556  60.155 
63.874  64.516  65.164  65.819  66.481 
70.591  71.300  72.017  72.741  73.472 


90  SOLUTION  OF  EQUATIONS 

Quadratic  Equation 

1.  Algebraic  Solution: 

*'  +J>*  +  q  =  o,        x=  -p/2  ±J  P\  _ry 

'    4 

ax*  +  bx  +  c  =  o,      *  -  -  b±  yj^^c 

2a 

2.  Solution  by  Slide  Rule: 

Let  the  four  scales  on  the  rule  be  A,  B,  C  and  D;  B  and  C 
being  on  the  runner,  and  B  and  A  the  squares  of  C  and  D. 

The  equation  having  been   reduced  to  the  form 

*2  4-  px  +g=o, 

set  the  indicator  at  q  on  D  and  move  the  runner  until 
the  sum  or  difference  of  ^Number  on  D  under  1  on  C)  and 
(Number  on  C  under  indicator)  is  equal  to  p.  Then  the  two 
numbers  are  the  required  roots. 

.  The  following  table  has  been  arranged  to  facilitate  the  find- 
ing of  the  positive  root  since  this  one  is  usually  of  more  import- 
ance than  the  negative. 

The  column  headed  "Limit  for  positive  root"  shows  whether 
the  positive  root  (c)  is  greater  or  less  than  the  limit,  which  is 
seen  to  be  a  number  either  known  or  easily  found.  It  is  best 
to  set  this  limit  on  (C)  under  the  indicator,  and  then  move  the 
runner  to  the  proper  setting  by  increasing  or  decreasing  the 
numbers  (on  the  runner  under  the  indicator)  according  as  c  is 
greater  or  less  than  the  limit. 

After  the  setting  is  made  which  satisfies  the  equation  in  the 
column  headed  "setting,"  the  positive  root  is  c,  that  is,  appears 
under  the  indicator. 

It  is  well  to  remember  that  if  4q  =  p2,  the  roots  are  equal 
and  if  p2  <  4q,  the  roots  are  imaginary. 

Form  Setting     Sign  of  Roots          Limit  for 

Larger     Smaller  Positive  Root 

x2  -f-  px  —  q  =  o      p  =  d  —  c       —       -|-        c  <  7/p  and  <C  |/  q 
x-  —  px  -j-  q  =o        p  =  c  +  d       +       ~h        c  <C  p,  d  <^  p 
x~  —  px  —  q  =  o      p  =  c  —  d        +  c  >  p 

c  and  d  are  the  two  roots;  d  =  number  on  D  under  1  on  C 
and  c  —  number  on  C  under  indicator. 

The  equation  xz  -j-  px  +  q  =  o  has  both  roots  negative  and 
p  =  c  +  d. 


SOLUTION  OF  EQUATIONS  91 

To  illustrate  the  method,  take  the  equation 

x2  +  0.33  x  —  5.33  =  0 

Trying  both  limits,  we  find  that  the  second  one  gives  the 
lower,  c  <C  2.31.  Since  q  =  5.33,  we  set  the  indicator  at  5.33 
on  D,  and  starting  with  2.31  under  the  indicator  and  moving 
the  runner  so  that  its  numbers  under  the  indicator  decrease,  we 
find  after  a  few  trials,  that 


( p  W  d  W  c  "I 

V0.33/      \2ASJ      \2.15J* 
Hence,    the  roots    are  +2. 15    and  —  2.48. 


Cubic  Equation 

1.  General  Form 

A  cubic  equation  of  the  form 

23  +  az2  +  bz  +  c  =  0  (1) 

is  best  solved  by  Horner's  method,  but  the  slide  rule  is  perhaps 
shorter  for  a  cubic  of  the  form 

x*  +  vx  +  q  =  0.  (2) 

Any  cubic  of  form  (1)  is  reduced  to  form  (2)  by  putting 
z  =  x  —  a/3. 

2.  Solution  by  Slide  Rule. 

The  solution  is  similar  to  that  for  the  quadratic.  Let  the 
equation  be  of  the  form  (2).  Set  the  indicator  at  q  on  D  and 
move  the  runner  until  the  sum  or  difference  of  (Number  on  B 
under  indicator)  and  (Number  on  D  under  1  on  C)  is  equal  to  p. 
Then  the  number  on  C  under  the  indicator  is  the  root. 

Form 

x'A  — px  —  q 
x'A  +  px  —  q 
.r3  —  px  +  Q 

b  =  number  on  B  under  indicator,  d  =  number  on  D  under 
1  on  C  and  c  =  number  on  C  under  indicator  =  root. 

In  solving  xs  —  px  +  q  =  0,  it  may  be  more  convenient  to 
first  find  the  negative  root  y.  The  limit  of  the  negative  root  is 
y^>  V  P  or  ^>  $  Q  and  the  setting  is  p  =  b  — d.  Then  the 
positive  roots  are  the  roots  of  the  quadratic  x2  —  yx  -f-  d  =  0, 
and  are  imaginary  if  7  <  f  4q. 


Setting 

Sign  of            Limit  for 
Roots              Positive  Root 

0 

p  = 

b 

—  d 

1  pos.  2  neg. 

c 

>V  P 

0 

p  = 

d 

-b 

1  pos.  2  imag. 

c 

<J&/~~     ' 
v    Q 

0 

p  = 

b 

+  d 

2  pos.  1  neg. 

c 

<VP 

92 


SOLUTION   OF  EQUATIONS 
Descartes'  Rule  of  Signs. 


An  equation  with  real  coefficients  has  no  more  real  positive 
roots  than  there  are  changes  of  sign  in  the  origonal  equation, 
and  no  more  real  negative  roots  than  there  are  changes  of  sign 
in  the  equation  when  —  x  is  put  for  x. 

Homer's  Method. 
To  find  the  positive  root  of  x3  -j-  ftx?  —  23x  —  70-0 


+   2 
5 


—  23 

+  35 

+  12 

60 


—  70 
4-60 


—  10.000 
7.371 


'   12 
5 

+  72.00 
1.71 

—  2.629000 
2.278497 

+  17.0 
.1 

73.71 
1.72 

—   .350503 
.306152 

.1 

+  75.4300 
.5199 

—   .044351 
.038300 

17.2 
.1 

75.9499 
.5208 

.006051 
5362 

+  17.30 
.03 

+  76.470 
.068 

689 

688 

17.33 
.03 

76.538 
.068 

• 

17.36 
.03 

76.60 

+  17. 

\ 

Explanation: — Write  down  the  coefficients  of  the  powers, 
putting  zero  for  any  power  that  is  wanting.  Thus,  for  the 
equation  x4  +  7x  —  £  =  0,  the  coefficients  would  be  written 

1  -f  0  +  0  +  7  —  9. 

If  the  integers  between  which  the  root  lies  are  known, 
multiply  the  first  coefficient  by  the  lower  integer,  place  the  pro- 
duct under  the  second  coefficient  and  add  algebraically.  As  in 
the  present  example,  the  root  lies  between  5  and  6.  Multiply- 
ing 1  by  5,  placing  the  product  5  under  the  second  coefficient 
-f  2  and  adding,  we  obtain  -f  7.  Multiplying  -f  7  by  5,  placing 
the  product  under  the  third  coefficient  —  23  and  adding,  we 
obtain  4-  12.  Repeating  this  process,  the  remainder  —  10  is 
obtained. 

If  the  two  numbers  between  which  the  root  lies  are  not 
known,  try  different  ones  until  two  are  found  which  give  re- 
mainders of  different  sign.  Then  the  root  lies  between  them. 
In  practice,  it  rarely  occurs  that  more  than  one  root  lies  witfrin 


SOLUTION   OF  EQUATIONS  93 

Horner's  Method — Continued 

the  interval  between  two  successive  integers,  so  that  it  is  usually 
sufficient  to  try  the  whole  numbers,  0,  1,  2,  3,  etc.,  or  0, — 1, — 2, 
etc.  In  the  example  above,  5  gives  the  remainder  —  10  while 
6  gives  -)-  80;  hence,  the  root  lies  between  5  and  6. 

If  the  remainder  is  zero,  the  number  is  an  exact  root  of  the 
equation. 

Continue  this  process  of  multiplying  by  the  root  number  and 
adding  the  product  algebraically  to  the  number  at  the  bottom 
of  the  next  column,  till  each  column  from  the  right  is  increased 
by  one  row,  as  indicated  by  the  broken  line. 

The   process  just  completed  for  the  coefficients  1-1-2  —  23 

—  70   is   to  be   repeated  for  the  coefficients   1  +  17  +  72  — 10. 
Trying  0.1  and  0.2,   we  find  that  the   first  gives   a    remainder 

—  2.629  and  the  second  a  remainder  -\-  5.088;    hence,    the   root 
to  one  decimal  is  5.1.     Applying  0.1  in  the  same  way  that  5  was  > 
we  arrive  at  another  set  of  coefficients  1  +  17.3  -f  75.43  —  2.629 
shown  just  below  the  second  broken  line. 

This  process  of  adding  successive  figures  to  the  root  may  be 
continued  until  zero  remainder  is  reached  or  any  required  degree 
of  approximation  is  obtained. 

If  five  or  six  decimal  places  are  required,  it  is  convenient  to 
find  only  the  first  two  or  three  by  the  above  process,  finding  the 
remainder  by  the  contracted  method.  In  this  case  we  will  begin 
the  contraction  after  having  found  the  second  place  in  the  root. 

From  the  coefficient  in  the  second  column  from  the  right, 
we  drop  one  figure  and  from  the  third  column  we  drop  two. 
The  fourth  column  we  drop  altogether,  so  that  our  coefficients 
appear  17  +  76.470  —  0.350503,  as  shown  below  the  third 
broken  line.  Applying  upon  these  coefficients  the  process  em- 
ployed at  the  beginning,  we  find  that  0.004  gives  a  negative  re- 
mainder and  0.005  a  positive  one,  and  hence,  that  4  is  the  next 
figure  in  the  root.  Continuing,  we  arrive  at  the  coefficients 
17  -f  76.606  — 0.044351.  By  dropping  the  last  figure  in  the 
second  column  and  the  two  in  the  third  we  have  only  the 
coefficients  76.60  and  0.044351  left.  Dividing  0.044351  by 
76.60  we  find  three  more  places,  so  that  the  root  is  5.134579, 
which  is  correct  to  five  places,  the  value  to  nine  places  being 
5.134578725. 

The  explanation  is  given  for  a  cubic  as  this  occurs  more 
frequently  in  engineering  problems,  but  it  is  plain  that  the  pro- 
cess applies  to  an  equation  of  any  degree. 


94  SOLUTION   OF  EQUATIONS 

Homer's  Method — Continued 

Homer's  method  is  greatly  facilitated  by  using  the  slide  rule 
for  performing  the  multiplications  and,  unless  great  accuracy  is 
required,  the  root  may  be  found  at  once,  remembering  any  num- 
ber which  will  reduce  the  last  remainder  to  zero  is  a  root.  For 
example,  setting  the  slide  at  5.13  and  multiplying  success- 
ively by  7.13  and  13.58,  we  obtain  the  small  remainder  —  0.4  as 
shown  below. 

1+2  —23.00      —70.0       15^,3' 

+  5.13      +36.58     +69.6 
+  7.13      +13.58       -   0.4 

It  will  usually  require  but  a  few  trials  to  locate  the  number 
which  will  give  the  smallest  remainder. 

Newton's  Method 
Let  a  =  first  approximation  to  root, 

b  =  (a  +  5)   =  second  approximation  to  root. 
Then  5  is  found  from 

«=_/(«> 
/(«) 
in  whichy'  («)  =  first  derivative  of  /(a). 

(b  +  5')   =  third  approximation  in  which 

,'-_/.<*> 
f'(b) 

This  process  is  carried  on  till  the  required  degree  of  approx- 
imation is  reached. 

By  Plotting  or  by  Use  of  Tables 

1.  By  plotting. 

Separate  equation  into  two  parts,  plot  to  scale   each  part 
and  find  value  of  variable  for  point  of  intersection. 
Example — To  find  root  of 

x'a  +  7x  —  sin  x  =  0 
Rewrite  in  form 

y  =  x5  +  7x  y  =  sin  x 

Plot  the  two  curves  and  find  point  of  intersection. 

2.  By  the  use  of  tables. 
Example — To  find  root  of 

a  —  tan  x  +  ex  =  0 

Find  from  table  value  of  ex  such  that  a  +  e*  is  equal  to 
value  of  tan  x  as  found  from  table  of  natural  tangents. 


fir, 


SERIES 


n 


rn-r 


1.    (a  4-  6)-  =  a-  +  na-»6  4-^T^  *"W  + 


2. 
3. 


4. 


1.1       .  1   1-3 


1.1.3.5 
2.4.6-8 


1.3-5 


zs 


1.3.5.7 


4 
X 


5.    e*  = 


6.    sin*-*      3!  +  5!      7, 


-  ^  2T      ,     X  X- 

7.    cosx  =  l--  +  --- 


INTEGRALS 


1.    I  udu  =  uv  —  I  vdu. 


l5 


Jx-rfz  =  |^p      16.  /  Vi 


=        V( 


e*. 


17 
M. 


a  -h 


5.  I  a:r  log  a  dx  =  a*. 

6.  I  sin  xdx  =  —  cos  x,  or  versin  x. 

7.  J  cos  xdx  =  sin  x,  or  —  coversin  x. 

8.  J  «  sin  xdx  —  sin  x—  x  cos  x. 

9.  J  x  cos  xdx  =  cos  x  -h  #sin  x. 

10  C   ***    =  -       ^  cosx         m  —  2  r 
'  J  smmx  ""       w  —  1    smm~!x      m  —  1  J 

11  (*   <fo  1          sinx         n  —  2  C  _& 
J  cosnx  ""  n  —  1  '  cos""1^      n  —  1  J  cosn 

X»  /*»  TJ- 

sin2mxc?x=  I    cos*mxdx  =  -• 
*/o  2 


13. 


14. 


-x     -  a 


2±a2zha2  log(x  -f  Vx2  ±  a2)]. 


NOTE:—  To  each   of  the  above  integrals   must  be  added    the 
constant  of  integration. 


INTEGRALS 


18'/v  f    =  log(x  +  v*T+"3i)  =  sinh~'  \ 

J     Vx*  —  a*  *' 

20.   f  -5^5=  -tan-i-. 
*/     a  "f~  x         fl  ^ 

81.    §  =  —  log  =  —  tanh"1 " 


22. 


23. 


_  _  1  log  fa 

«    V 


24. 


25. 


J. 


dx 


V(x2  dt  a2)8      a2  V x2  ±  a2 

26.    f-       ^       •=         -^ 

J   V(a2  —  x2)8      a2  Va2  —  x2 

NOTE: — To  each  of  the  above  integrals  must  be  added  the 
constant  of  integration. 


'98  INTEGRALS 

DEFINITE   INTEGRALS 

Approximate  Evaluation  of  Definite  Integrals. 

b 

To  evalute  the  integral  J  f(x)  dx  divide  (b  —  a)  into  n  equal 
a 

parts,    }     a  =  h,  and  for  the  values  of 

n 

x  =  a,  x  =  a  +  h,  x  =  a  +  2h.  .  .  ., 

find  the  values  of 

/  (x)  =  To,  Yi,  rlt  .  . 
b 

J    f  (x)  dx  -  h  [%  Fo  +  Fi  +  Yt  +....+  Yn  -i  +  K  Yn] 


-  ff   (f  (^  -/  (a)  ]  ;+  B£  (f™  (b)  -/>"  (a)  ]  .... 


-  V. 


However,  it  is  not  usual  to  consider  the  terms  containing 
the  coefficients  A  and  B. 

SIMPSON'S  RULE 
b 

J  /  (*)  dx  =-    3    [T0  +  2  (F2  +    F4  +   +  rn  _  2) 

a 

+  4(r,  +  Y,  +  +  r^_,)  +  Yn] 

To  apply  this  rule  n  must  be  an  even  number. 

PARABOLIC  RULE 

6 


J 


EQUATIONS   OF   CURVES 


Logarithmic  Spiral 

r  =  ae  md 

m  =  ct  n  a  =  constant 


Catenary 


s  =  length  of  curve  measured 
from  vertex. 

*  =  c  tan  T  =  c  sinh  -2- 

c 

c  =  distance  of  vertex  from 
x  axis. 
H 


H  =  horizontal  tension  at  vertex, 
e  =  weight  per  unit  length. 
y*  =  c2  +  s2 


)  =  c  cosh  3 
c 


c  sinh 

c 


100  PROBABLE   ERROR 


PROBABLE    ERROR 

Weighted  Observations.  —  If  all  the  observations  are  con- 
sidered not  to  be  of  equal  value,  they  are  to  be  weighted  to 
such  an  extent  that  the  doubtful  ones  will  be  reduced  to  their 
proper  values  as  regards  the  probable  error. 

Any  observation  considered  as  being  of  maximum  reli- 
ability is  given  a  certain  weight,  as  10.  The  other  observa- 
tions are  weighted  accordingly;  as,  for  example,  an  observation 
taken  under  very  unfavorable  circumstances  and  hence  quite 
doubtful  might  be  given  a  weight  of  2  or  even  1,  whereas  an 
observation  considered  as  being  only  slightly  deficient  in  reli- 
ability would  be  weighted  9. 

The  weighted  mean  =  ^  w  R 

2  w 

in  which  w  is  the  weight  of  any  observation  and  R  is  the  value 
of  that  observation. 
Then,  if 

d  =  difference  between  R  and  the  weighted  mean 

n  =  number  of  observations 

e  =  probable  error  of  any  single  observation]of  weight  unity 

E  =  probable  error  of  the  weighted  mean 


0.67449 


-w- 
\  n  —  1 


=  0.67449    _, 

n — 1)  2  w 


[S  wdr 
V  Z* 


Unweighted  Observations.     If  all  the  observations  are  taken 
to  be  of  equal  reliability,  the  above  equations  reduce  to 

e  =C  v'~Z"d2and  E=       e    --  =  c  V  2  & 
V  n 


PROBABLE " ERROR 


101 


Values  of  C'and  c.> 
n          C          c 


c 


2 

0.6745 

0.4769 

21 

0.150$ 

0.0329 

40 

0.10SO 

0.0171 

3 

.4769 

.2754 

22 

.1472 

.0314 

41 

.1066 

.0167 

4 

.3894 

.1947 

23 

.1438 

.0300 

42 

.1053 

.0163 

5 

.3372 

.1508 

24 

.1406 

.0287 

43 

.1041 

.0159 

6 

.3016 

.1231 

25 

.1377 

.0275 

44 

.1029 

.0155 

7 

.2754 

.1041 

26 

.1349 

.0265 

45 

.1017 

.0152 

8 

.2549 

.0901 

27 

.1323 

.0255 

46 

.1005 

.0148 

9 

.2385 

.0795 

28 

.1298 

.0245 

47 

.0994 

.0145 

10 

.2248 

.0711 

29 

.1275 

.0237 

48 

.0984 

.0142 

11 

.2133 

.0643 

30 

.1252 

.0229 

49 

.0974 

.0139 

12 

.2034 

.0587 

31 

.1231 

.0221 

50 

.0964 

.0136 

13 

.1947 

.0540 

32 

.1211 

.0214 

55 

.0918 

.0124 

14 

.1871 

.0500 

33 

.1192 

.0208 

60 

.0878 

.0113 

15 

.1803 

.0465 

34 

.1174 

.0201 

65 

.0843 

.0105 

16 

.1742 

.0435 

35 

.1157 

.0196 

70 

.0812 

.0097 

17 

.1686 

.0409 

36 

.1140 

.0190 

75 

.0784 

.0091 

18 

.1636 

.0386 

37 

.1124 

.0185 

80 

.0759 

.0085 

19 

.1590 

.0365 

38 

.1109 

.0180 

90 

.0715 

.0075 

20 

1547 

.0346 

39 

.1094 

.0175 

100 

.0678 

.0068 

SECTIONS  AND  SOLIDS 
Guldin's  Rules. 

I.  If  a  plane  area  revolve  about  an  axis  external  to  itself 
through  any  assigned  angle,  the  volume  of  the  solid  generated 
will  be  equal  to  a  prism  whose  base  is  the  revolving  area,  and 
whose  altitude  is  the  length  of  the  path  described  by  the  center 
of  mass  of  the  area. 

II.  If  the  arc  of  a  plane  curve  revolve  about  an  external 
axis  in  its  own  plane  through  any  assigned  angle,  the  area  of 
the  surface  generated  will  be  equal  to  that  of  a  rectangle,  one 
side  of  which  is  the  length  of  the  revolving  curve,  and  the  other 
the  length  of  the  path  described  by  its  center  of  mass. 

The  Prismoidal  Formula 
Let   A  =  area  of  the  base  of  a  prism, 
At,   A2,  Am  =the  two  end  and  middle  areas  of  a 

prismoid,  or  of  any  of  its  elementary  solids, 
h  =  altitude  of  the  prismoid  or  elementary  solid, 
V  =  its  volume 
V  =  ft/6  (A,  +  4  Am  +  A,) 
For  a  prism  Al  =  Am  =  A2  =  A, 

V  =  ft/6  X  6  A  =  h  A. 
For  a  wedge  with  parallel  ends;  A2  =  0,  Am  =  %  At, 

V  =  ft/6  (  A,  +  2  A,  )  =   h  A- 
For  a  cone  or  pyramid  A,  =  0,  Am  =  %  Ax, 


ft/6  (A,  +  A,) 


o 


102  SECTIONS   AND  SOLIDS 

Momencs  of  Inertia. 

The  moment  of  inertia  of  a  body  about  a  given  axis  is  the 
summation  for  all  the  particles. of  the  body  of  the  product  mass 
of  particle  X  squire  of  its  distance  from  the  given  axis;  or  sym- 
bolically, if  dm  is  the  elementary  mass  and  r  its  distance  from 
the  axis, 


Moment  of  Inertia  =    j    r2   dm', 


the  summation  or  integration  being  taken  so  as  to  include  the 
total  mass  of  the  body. 

The  fundamental  properties  of  moments  of  inertia  are  the 
following: 

1.  The  moment  of  inertia  of  a  body  about  a  given  axis  is  the 
sum  of  the  moments  of  inertia  of  its  parts  taken  about  the  given 
axis. 

2.  If  /  is  the  moment  of  inertia  of  a  body  about  an  axis 
through  its  center  of  mass  and  /'  its  moment  of  inertia  about 
any  parallel  axis, 

lf  =  I+r*M; 

where  r  is  the  distance  between  the  two  axes  and  M  is  the  mass 
of  the  body. 

Axes  lying  in  the  surface  of  a  thin  lamina  or  plane  section 
are  known  as  equatorial  axes;  axes  perpendicular  to  the  lamina 
as  polar  axes. 

3.  For    a   thin    lamina    or    plane  section,    the  sum  of  the 
moments  of  inertia  about  any  two  equatorial  axes  at  rightangles 
is  equal  to  the  moment  of  inertia  about  the  polar  axis  passing 
through  their  intersection. 

Area,  Center  of  Mass,  Moments  of  Inertia,  Radius  of  Gyration 
and  Section  Modulus  of  Sections 

E  =  area  of  section. 

G  =  radius  of  gyration  for  moment  of  inertia  I. 

I  =  moment  of  inertia  about  an  axis  shown  by  dotted  line. 

J  =  moment  of  inertia  about  an  axis  perpendicular  to  plane 
of  section. 

Except  where  indicated  by  subscripts,  I  and  J  are  taken  about 
axes  through  the  center  of  mass  of  the  section  or  solid. 

If  K  =  moment  of  inertia  about  the  equatorial  axis  through 
the  center  of  mass  and  perpendicular  to  the  axis  for  I,  then 
J  =  I  +  K. 

S  =  section  modulus  for  I. 

x  locates  the  center  of  mass. 


SECTIONS  AND  SOLIDS 


103 


Area,  Center  of  Mass,  Moments  of  Inertia,  Radius  of  Gyration 
and  Section  Modulus  of  Sections. 


L 


E^bd-fyd,*    f*£(M*-kd') 


/z 


X  = 


J=E(b*d  +b<f-b*d,  - 


6d 


ai 


=  0. 


/=         =  0.  049  d 


104 


SECTIONS   AND   SOLIDS 


Area,  Center  of  Mass,  Moments  of  Inertia,  Radius  of  Gyration  and  Sec- 
tion Modulus  of  Sections. 


Q 


/=    *<•*„  d*=0.007d* 

32    . 


2 

r*a 


u  /J  measured  m  radians. 


a3      tr^irfu 


3  £ 


ETZ/       2    sin3Uco$U       \ 

T-  _J_  ( i  i — _  )  —  P  y2 

"    ^    ^         J    U-sinUcosUf       L  * 

U  is  measured  in  radian-s. 


SECTIONS   AND   SOLIDS 


105 


Area,  Center  of  Mass,  Moments  of  Inertia,  Radius  of  Gyration  and  Sec- 
tion Modulus  of  Sections. 


r- 


REGULAR    POLYGONS 

R  =  radius  of  circumscribed 
r  =  radius  of  inscribed  circle 
a  -  2V  R2  —  r2  =  side; 
n  =  number  sides 
«  =  180°/n 


=  7= 


'" .  I6a3b 


d 

~~     >    J~ 


xld 


106 


SECTIONS   AND   SOLIDS 


Area,  Center  of  Mass,  Moments  of  Inertia,  Radius  of  Gyration 
and  Section  Modulus  of  Sections. 


=bd-hu,   /= 

d 

2  5- 


bd3-uh3 


6d 


^U-. 


~h 


d 


6d 


^i~\?i.       E=ds+ht 

JL          sd*+ht2 
b-Q    *=—E '    y  =  d~X 


.6. 

—A 

.6. 

JL 

-b- 

SECTIONS   AND   SOLIDS 


107 


Surface,   Volume,   Center   of  Mass  and   Moments  of  Inertia  of 

Solids 

B  =  area  of  base. 

I  =  moment  of  inertia  about  axis  I-I. 
J  =  moment  of  inertia  about  axis  J-J. 
I  and  J  are  for  axes  through  the  center  of  mass. 
L  =  surface  of  solid^exclusive  of  base  or  bases. 
M  =  mass. 
V  =  volume. 
x  locates  the  center  of  mass 


-5  £ 


V=Zabh 


108 


SECTIONS  AND   SOLIDS 


Surface,  Volume,  Center  of  Mass  and  Moments  of  Inertia  of  Solids 


SPHERE 
Radius  r 


.a,  a. 


1= 


^o 


Circular  Crott  Sectio 


Elliptical  Cross  Section.  Axes 
and  zb,    b  axts  parallel  to  JJ. 


SECTIONS  AND  SOLIDS  109 

Coefficients  of  Strength  and  Deflection  for  Steel  Shapes 

In  the  tables  the  "coefficient  of  strength"  is  the  safe  load 
in  pounds  uniformly  distributed,  including  weight  of  beam,  for  a 
steel  beam  one  foot  long.  Hence,  for  any  other  span  the  safe 
uniform  load  in  pounds  is  obtained  by  dividing  the  proper 
coefficient  by  the  span  in  feet.  Reduce  other  than  uniform 
loads  to  equivalent  uniform  load  by  factor  X,  page  130  et  seq. 

The  "  coefficient  of  deflection"  is  the  deflection- in  inches  of 
a  steel  beam  one  foot  long  and  loaded  with  1000  pounds.  For 
any  other  load  and  span,  obtain  the  deflection  in  inches  by 
multiplying  the  proper  coefficient  by  the  cube  of  the  span  in 
feet  and  the  number  of  1000-pound  units  in  the  load. 
Properties  of  Simple  and  Compound  Section  s 

Usually,  only  the  moments  of  inertia  for  the  axis  1—1  and 
2 — 2  are  required,  but  in  case  this  quantity  for  some  other  axis 
is  wanted  it  may  be  found  from  the  tables  as  follows. 

1.  For  an  axis  parallel  to  1 — 1  or  2—2,  find  according  to  2 
page  102. 

2.  For  an  axis  inclined  to  1 — 1,  find  first  the  moment  of 
inertia  K  about  an  axis  through  the  center  of  mass  (gravity)  of 
the  shape  and  parallel  to  the  required  axis.       Then  apply    2, 
page  102.     For  I-beams,  bulb  beams,  channels  and  T-bars 

K  =  I  cos2  |8  +  1'  sin  2  ft 

in  which  j8  is  the  angle  which  the  inclined  axis  makes  with  the 
axis  1 — 1     For  angles  and  Z-bars, 

K  =  I"  cos2  /3  +  I'"  sin2  j8, 

i"  =  Ar"\    r  =  i  +  r  -i" 

and  /3  is  the  angle  which  the  axis  for  K  makes  with  the  axis  3—3. 

For  compound  steel  sections  formed  by  combining  simple 
shapes  such  as  plates,  angles,  etc.,  the  moment  of  inertia  about 
an  axis  through  its  center  of  inasH  will  usually  be  required. 
The  center  of  mass  for  symmetrical  sections  is  at  the  center. 
For  an  unsymmetrical  section,  assume  any  convenient  line  as  an 
axis,  multiply  the  area  of  each  simple  shape  by  the  distance  of 
its  center  of  mass  from  this  axis  and  divide  the  sum  of  these 
products  by  the  sum  of  the  component  areas.  The  quotient 
is  the  distance  of  the  center  of  mass  of  the  compound  section 
from  the  assumed  axis. 

The  section  modulus  is  the  moment  of  inertia  divided  by  the 
distance  of  the  most  remote  point  of  the  section  from  the  axis 
about  which  the  moment  of  inertia  is  taken. 

The  radius  of  gyration  is  the  square  root  of  the  quotient 
obtained  by  dividing  the  moment  of  inertia  by  the.  area. 


110 


SECTIONS  AND   SOLIDS 
Properties  of  Cambria  Standard  I-Beams. 


1 

2 

3 

4 

6 

6 

7 

8 

9 

10 

11 

Kadius 

Radius 

Section 

T 

Beam. 

Weight 
Foot 

Area 
of 

Section. 

Thick- 
ness of 
Web. 

Width 
of 

Flange. 

Moment 
of 
Inertia 
Axis  1-1. 

Section 
Modulus 
Axis  1-1. 

of 
GTTE- 
tion 
Axis 

Moment 
of 
Inertia 
Axis  24 

of 

Gyra- 
tion 

Alls 

Number. 

1-1. 

2-2. 

d 

A 

t 

b 

I 

S          r 

I' 

r' 

Inches. 

Pounds. 

Sq.Ins. 

Inch. 

Inches. 

Inches.* 

Inchest  Inches. 

Inches.* 

Inch. 

B    5 

3 

6.50 

1.63 

.17 

2.33 

2.5 

1.7 

1.23 

.46 

.53 

«i 

tt 

6.5O 

1.91 

.26 

2.42 

2.7 

1.8    .19 

.53 

.52 

H 

tt 

7.50 

2.21 

.36 

2.52 

2.9 

1.9 

.15 

.60 

.52 

B    9 

4 

7.50 

2.21 

.19 

2.66 

6.0 

3.0 

.64 

.77 

.59 

14 

tt 

8.50 

2.5O 

.26 

2.73 

6.4       3.2    .59 

.85 

.58 

(« 

tt 

9.50 

2.79 

.34 

2.81 

6.7 

3.4    .54 

.93 

.58 

11 

" 

1O.5O 

3.09 

.41 

2.88 

7.1 

3.6 

.52 

1.01 

.57 

BIS 

5 

9.75 

2.87 

.21 

3.00 

12.1 

4.8 

2.05 

1.23 

.65 

M 

tt 

12.25 

3.6O 

.36 

3.15 

13.6 

6.4  1.94 

1.45 

.63 

It 

tt 

14.75 

4.34 

.50 

3.29 

15.1 

6.1 

1.87 

1.70 

.63 

B17 

6 

12.25 

3.61 

.23 

3.33 

21.8 

7.3 

2.46 

1.85 

.72 

(i 

tt 

14.75 

4.34 

.35 

3.45 

24.0 

8.0i2.35 

2.09 

.69 

it 

it 

17.25 

5.O7 

.47 

3.57 

26.2 

8.7 

2.27 

2.36 

.68 

B21 

7 

15.OO 

4.42 

.25 

3.66 

36.2 

10.4 

2.86 

2.67 

.78 

ii 

17.50 

5.15 

.35 

3.76 

39.2 

11.  2  j  2.76 

2.94 

.76 

,i 

tt 

20.00 

5.88 

.46 

3.87 

42.2 

12.1 

2.68 

3.24 

.74 

B26 

8 

18.00 

5.33 

.27 

4.00 

56.9 

14.2 

3.27 

3.78 

.84 

tt 

tt 

20.25 

5.96 

.35 

4.08 

60.2 

15.O3.18 

4.04 

.82 

(< 

** 

22.75 

6.69 

.44 

4.17 

64.1 

16.0  3.1  0 

4.36 

.81 

II 

" 

25.25 

7.43 

.53 

4.26 

68.O 

17.0 

3.O3 

4.71 

.80 

B29 

9 

21.00 

6.31 

.29 

4.33 

84.9 

18.9 

3.67 

5.16 

.90 

u 

25.0O 

7.35 

.41 

4.45 

91.9 

2O.43.54j    6.65 

.88 

it 

" 

30.00 

8.82 

.57 

4.61 

1O1.9 

22.6  13.40 

6.42 

.85 

« 

«t 

35.OO 

10.29 

.73 

4.77 

111.8 

24.8 

3.30 

7.31 

.84 

B33 

10 

25.00 

7.37 

.31 

4.66 

122.1 

24.4 

4.07 

6.89 

.97 

it 

30.00 

8.82 

.45 

4.8O 

134.2 

26.8 

3.9O 

7.65 

.93 

it 

*' 

35.00 

10.29 

.60 

4.95 

146.4 

29.3!  3.77 

8.52 

.91 

t< 

tt 

40.0O 

11.76 

.75 

5.10 

158.7 

31.7 

8.67 

9.50 

.90 

B41 

12 

31.50 

9.26 

.35 

5.00 

215.8 

36.0 

4.83 

9.50 

1.01 

<» 

tt 

35.00 

10.29 

.44 

5.O9 

228.3 

38.O  4.71 

10.07 

.99 

it 

tt 

40.00 

11.76 

.56 

5.21 

245.9 

41.0 

4.57 

10.95 

.96 

B53 

15 

42.00 

12.48 

.41 

6.50 

441.8 

58.9 

5.96 

14.62 

1.08 

tt 

ii 

45.00 
60.00 

13.24 
14.71 

.46 
.56 

5.55 
6.65 

455.8 
483.4 

6O.8  5.87 
64.5  5.73 

15.09  1.07 
16.O4  1.O4 

*  tt 
ti 

tt 
«t 

55.00  16.18 
60.00  17.65 

.66 
.75 

5.75     611.01    68.1  6.62 
5.84    5S8.61    71.8!5.52! 

17.06  1.O3 
18.17  1.01 

SECTIONS' AND  SOLIDS 
Properties  of  Cambria  Standard  I-Beams. 


ill 


12 

13    |    14 

15     1     16 

1 

Increase  of 
Thickness 
of  Web  for 
each  Pound 
Increase 
in  Weight. 

Coefficient  of  Strength. 

Coefficient  of  Deflection. 

Section 
Number. 

B  5 

ti 
t( 

For  Fibre  Stress 
of  16  000  Pounds 
per  Square  Inch 
for 
Buildings. 

For  Fibre  Stress 
of  12  500  Pounds 
per  Square  Inch 
for 
Bridges, 

Uniform 
Load. 

Center 
Load. 

f 

F 

F' 

N 

N' 

.098 

17650 
1914O 
20710 

13790 
14950 
16180 

.OOO31253 
.OOO28827 
.00026644 

.00050006 
.00046124 
.OOO4263O 

.074 

31810 
3389O 
3598O 
3807O 

24850 
2648O 
2811O 
29750 

.00013009 
.OOO122O9 
.OOO115OO 
.OOO  10868 

.00020815 
.OO019535 
.OOO184OO 
.OOO17389 

B  0 
it 
« 

YC 

.050 

61590 
581OO 
64630 

40300 
45390 
50490 

.OOO06417 
.O0005698 
.00005122 

.00010267 
.OOOO9117 
.OOO08195 

B13 
ti 

.049 

77460 
8527O 
9311O 

60520 
6661O 
72740 

.00003561 
.OOO03235 
.00002963 

.00005698 
.OOOO  5  177 
.00004741 

B17 

C( 

.042 

110410 
1194OO 
12856O 

86260 
9329O 
1OO43O 

.00002142 
.OOOO  1980 
.00001839 

.OOOO3427 
.OOO03168 
.00002943 

B21 

M 

1  it 

.037 

151660 
16O51O 
17O97O 
181480 

118490 
12540O 
13357O 
141740 

.OOOO  1364 
.OOOO  1289 
.OOOO121O 
.00001140 

.00002183 
.OOOO2062 
.OOOO  193  6 
.00001825 

B25 

C( 

{( 

.033 

201300 
21793O 
24146O 
264990 

157260 
170260 
18864O 
207020 

.O0000914 
.OOOOO844 
.OOOOO762 
.OO000694 

.00001462 
.OOO0135O 
.OOOO121  9 
.O0001110 

B29 

M 

tt 

.029 

260470 
28625O 
31239O 
33853O 

2O350O 
22363O 
244O50 
26448O 

.OO000635 
.OOOOO578 
.OO000530 
.OO000489 

.00001017 
.OOOO0925 
.00000848 
.OOOO0782 

B33 

M 
«« 

.025 

383670 
4058OO 
437170 

29974O 
317030 
341  54O 

.OOOOO36O 
.OOOOO34O 
.O0000316 

.OOOO0575 
.OOOOO544 
.OOOO0505 

B41 
el 

.020 

628270 
6483  1O 
687530 
72674O 
765960 

490840 
5O649O 
63713O 
567770 
69841O 

.00000176 
.OOOOO17O 
.OOOOO1  61 
.OOOOO  152 
.OOOOO  144 

.00000281 
.OOOO0272 
.OOOOO257 
.00000243 
.OOOOO231 

B53 
it 

«i 
i< 

112 


SECTIONS  AND   SOLIDS 


Properties  of  Cambria  Standard  I-B  earns. 


1 

2 

3 

4 

5 

6 

7 

8     1    9    |     10 

11 

Radius 

Radius 

Section 
Number. 

Depth 
of 
Beam. 

Weight 

J*r 
Foot. 

Area 
of 

Section. 

Thick- 
ness of 
Web. 

Width 
of 
Flange. 

Moment 
of 
Inertia 
Axis  1-1. 

Section 
Modulus 
Axis  1-1. 

of 
Gyra- 
tion 
Ails 
1-1. 

Moment 
of 
Inertia 
Aiis2-2. 

of 
Gjra- 
tion 
Axis 
2-2. 

d 

A 

t 

b 

I 

S 

r 

I' 

r' 

Inches. 

Pounds. 

Sq.Ins. 

Inch. 

Inches, 

Inches.* 

Inchest 

Inches. 

Inches.* 

Inch. 

B  65 

18 

65.0 

15.93 

.46 

6.00 

795.6 

88.4 

7.07 

81.10ll.16 

it 

tt 

60.O 

17.65 

.56 

6.1O 

841.8 

93.5  6.91  22.38  1.13 

cc 

tt 

65.O 

19.12 

.64 

6.18 

881.5 

97.9  6.79  23.47 

1.11 

tt 

tt 

70.O 

20.59 

.72 

6.26 

921.2 

102.4  6.69  24.62 

1.09 

B   73 

20 

65.0 

19.08 

.50 

e.25'1  169.5  117.0<7.83'27.86'1.21 

tt 

tt 

70.O 

2O.59 

.58 

6.33  1219.8  122.0  7.70  29.04  1.19 

tt 

tt 

75.0 

22.06 

.65 

6.4O  1268.8  126.9  7.58  30.25  1.17 

B   89 

24 

80.O 

23.32 

.50 

7.00  2087.2  173.9  9.46  42.86 

1.36 

tt 

85.O 

25.00 

.57 

7.07  2167.8 

180.7  9.31  44.35  1.33 

tt 
tt 

tt 

M 

90.0    26.47 
95.0    27.94 

.63 
.69 

7.13  2238.4H86.5  9.2O  45.7O  1.31 
7.19  23O9.O  192.4  9.09  47.10;  1.3O 

tt 

tt 

100.O 

29.41 

.75 

7.25  2379.6  198.3  8.99  48.55'  1.28 

Properties  of  Cambria  Bulb  Beams. 


ntL 


3.  p 

1 

/        1l            ^*^i 

1 

fl~  -J  -.         .^.—  ^^m».^M  W 

JL 

1             2 

3 

4 

5 

6      |      7 

8 

9 

Depth 

Area 

Thickness 

Width 

Moment  of    Section 

Radius  of 

Section 

of 
Beam. 

Weight 

of 

Section. 

ef 
Web. 

of 
Flange. 

Inertia 
Axis  1-1. 

Modulus 
Axis  1-1. 

Gyration 
Axis  1-1. 

Sumber. 

Foot. 

d 

A 

t 

b 

I 

8 

r 

Inches. 

Pounds. 

Sq.  Ins. 

Inch. 

Inches. 

Inches.* 

Inches.3 

Inches. 

B173 

M 

6 

14.0 
15.3 
18.4 

4.11 
4.48 
5.42 

8 

i 

21.52 
22.73 
25.72 

6.12 
6.55 
7.59 

2.29 
2.25 
2.18 

SECTIONS  AND   SOLIDS 


113 


Properties  of  Cambria  Standard  I-Beams. 


12 

13          I          14          |            15            I           16 

1 

Increase  of 
Thickness 
of  Web  for 
each  Pound 
Increase 
in  Weight. 

Coefficient  of  Strength. 

Coefficient  of  Deflection. 

Section 
Number. 

For  Fibre  Stress 
of  16  000  Pounds 
per  Square  Inch 
for 
Buildings. 

For  Fibre  Stress 
of  12  500  Pounds 
per  Square  Inch 
for 
Bridges. 

Uniform 
Load. 

Center 
Load, 

f 
.016 

F 

Jj 

N 

N' 

942880 
997680 
104474O 
1091800 

736620 
77944O 
81620O 
85297O 

.00000098 
.00000092 
.O0000088 
.00000084 

.00000156 
.OOOOO  148 
.OOOOO  141 
.OOOO0136 

B«66 

ti 
«i 

.015 

1247490 
1301110 
1353400 

974600 
1O16490 
105734O 

.00000066 
.OOOOOO64 
.00000061 

.00000106 
.OOOOO102 
.00000098 

Bu73 

N 

.0123 

1866310 
192695O 
19897OO 
2062440 
2115190 

1449460 
1505430 
1554450 
1603470 
1652490 

.00000037 
.00000036 
.OOOOO036 
.OOOOO034 
.OOOO0033 

.00000060 
.OOOOOO57 
.00000056 

.OCOOO054 
.00000052 

B   89 

ct 

II 
(1 
(( 

Properties  of  Cambria  Bulb  Beams. 


^Q~--' 


10 

11 

12 

13 

14        |        15 

1 

Distance  of 
Center  of 
Grarity  from 
Outside  of 
Flange. 

Increase  of 
Thickness  of 
Web  for  each 
Lb.  Increase 
in  Weight. 

Coef.  of  Strength. 

Coef.  of  Deflection. 

Section 
Number. 

Fibre  Stress 
16  000  Pounds 
per  Sq.  Inch 
for  Buildings. 

Fibre  Stress 

12  500  Pounds 
per  Sq.  Inch 
for  Bridges. 

Uniform  Load. 

Center  Load. 

X 

Inches. 

f 

F 

F 

N 

N' 

2.49 
2,53 
2.61 

.049 
it 

65320 
6986O 
80930 

5103O 
5458O 
63230 

.OOOO361 
.O000341 
.0000302 

.OOO0577 
.0000546 

.0000483 

B173 
H 

114 


SECTIONS  AND   SOLIDS 
Properties  of  Cambria  Standard  Channels. 


-It- 


1 

2          3 

4 

6 

6    !     7 

8 

9 

1O 

11 

12 

Radius 

Radius 

Thick- 

Moment ^H00 

of 

Moment 

Section 

of 

Soction 
Num- 

Depth 
of 

Channel. 

Weight 
per 
Foot. 

Area 
of 
Section. 

ness 
of 
Web. 

Width 
of 
Flange. 

of 
Inertia 
Axis  1-1. 

Mod- 
ulus 
Axis 
1-1. 

Gyra- 
tion 
Axis 
1-1 

of 
Inertia 
Axis  2-2. 

Mod- 
ulus 
Axis 
2-2. 

Gyra- 
tion 
Axis 
2-2. 

ber. 

(1 

A 

t 

b 

I 

S 

p 

1' 

S' 

T* 

Inches. 

Pounds.  |  Sq.  Ins. 

Inch,  i  Inches.  Inches.4 

Ins.3 

Inches, 

Inches.* 

Ins.3 

Inch. 

C    5 

3 

4.0oi    1.19    .17  1.41 

1.6 

1.1 

1.17 

.20 

.21 

.41 

«« 

tt 

6.00;    1.47    .26!  1.50 

1.8 

1.2 

1.12 

.26 

.24 

.41 

44 

u 

6.OO 

1.76 

.3611.60 

2.1 

1.4 

1.08 

.31 

.27 

.42 

C    9 

4 

-  6.25 

1.55 

.18  1.68 

3.8 

1.9 

1.56 

.32 

.29 

.45 

6.25 

1.84    .25 

1.65 

4.21   2..1 

1.51 

.38 

.32 

.45 

44 

tl 

7.26 

2.13    .33 

1.73 

4.6    2.3 

1.46 

.44 

.35 

-.46 

C13 

5 

6.50 

1.95    .19 

1.75 

7.4    3.0 

1.95 

.48 

.38 

.50 

tt 

9.00 

2.65 

.33 

1.89 

8.9     3.5 

1.83 

.64 

.46 

.49 

44 

44 

11.50 

3.38 

.48  2.O4 

10.4    4.2 

1.75 

.82 

.54 

.49 

C17 

6 

8.00    2.38    .20  1.92 

13.O 

4.32.34 

.70 

.50 

.54 

44      10.50J    3.09 

.32  2.04 

15.1 

6.O 

2.21 

.88 

.57 

.63 

" 

44      13.OOI    3.82 

.44  2.16 

17.3 

5.82.13 

1.07 

.65 

.63 

it 

44      15.50]    4.56 

.56  2.28 

19.5 

6.5(2.O7 

1.28 

.74 

.53 

C21 

7 

9.75     2.85 

.21 

2.O9 

21.1 

6.0  2.72 

.98 

.63 

.59 

12.25;    3.60 

.32 

2.20 

24.2 

6.9 

2.59     1.19 

.71 

.57 

« 

tt 

14.75!    4.34 

.42 

2.30 

7.8  2.56  i    1.40 

.79 

.57 

»» 

44    i  17.25 

6.O7 

.63 

2.41 

3O*2 

8.62.44i    1.62 

.87 

.66 

44 

44    J19.75 

6.81 

.63 

2.51 

33.2 

9.5  2.39     1.85 

.96 

.56 

C25 

8 

11.25 

3.35 

.22 

2.26 

32.3 

8.1 

3.10 

1.33 

.79 

.63 

tt 

13.75 

4.04 

.31 

2.35 

36.0 

9.0 

2.98 

1.551   .87 

.62 

tt 

it 

16.25 

4.78 

.40  2.44 

39.9  10.0  2.89 

1.78    .96 

.61 

i< 

it 

18.75 

5.51 

.49  2.53 

43.8  11.  0 

2.82     2.01  1.02 

.60 

tt 

44 

21.25     6.25 

.58  2.62 

47.8,11.92.76    2.251.11 

.60 

C29 

9 

13.25    3.89 

.23  2.43 

47.3!  10.5 

3.49     1.77    .97 

.67 

15.0O 

4.41 

.29 

2.49 

5O.9  11.3  3.40     1.951.03 

.66 

tt 

it 

20.00 

6.88 

.45  2.66 

60.8  13.5 

3.21 

2.45  1.19 

.65 

tt 

tt 

25.00 

7.35    .6112.81 

70.7  15.7 

3.10    2.98  1.36 

.64 

C33 

10 

15.00 

4.46     .242.60 

66.9  13.4 

3.87    2.3O  1.17 

.72 

20.00 

5.88     .3812.74 

78.7:15.7 

3.66^    2.85  1.34 

.70 

tt 

tt' 

25.00     7.35     .5312.89 

91.0  18.2 

3.521   3.40  1.50    .68 

i« 

tt 

30.OOJ    8.82 

.68'3.O4 

103.220.6 

3.42     3.99  1.67 

.67 

" 

44 

35.OO  10.29 

.8213.18 

115.523.1 

3.35 

4.66  1.87 

.67 

C41 

12 

20.50     6.03 

.28  2.94 

128.1  21.4  4.61 

3.91 

1.75 

.81 

25.OO     7.35 

.39  3.05 

144.0  24.0  4.43 

4.53  1.91 

.78 

tt 

it 

30.00     8.82 

.51 

3.17 

161.626.9 

4.28 

6.212.09    .77 

tt 

it 

35.00  1O.29 

.64 

3.3O 

179.3  29.9 

4.17 

6.902.27    .76 

tt 

»« 

40.00  11.76 

.76  3.42 

196.9,32.84.09 

6.63  2.46    .75 

Co  3 

15 

33.00 

9.90 

.403.40  312.641.7 

5.62 

8.23  3.16 

.91 

44     35.OO  1O.29 

.43 

3.43  319.942.7 

5.57     8.48  3.22 

.91 

/  tt 

||     40.0011.76 

.52 

3.52  347.5  46.3 

6.44     9.39  3.43 

.89 

t» 

45.00.13,24 

.62:3.62  375.1  50.O 

5.32  1O.29  3.63 

.88 

44 

ti 

50.0OJ  14.71 
55.00116.18 

.72  3.72  402.7  53.7,5.23,  11.22  3.85 
.82  3.82  430.2:57.4  5.16112.19  4.07 

.87 
.87 

SECTIONS  AND   SOLIDS 


115- 


Properties  of  Cambria  Standard  Channels. 


-It 


13      14 

15 

16 

17        18 

1 

Distance 

Increase  of 

Coef.  of  Strength. 

Coef.  of  Deflection. 

of  Center 
of  Gravity 
from 

Thickness  of  Fibre  Stress 
Web  for  each  16  000  Pounds 

1  Fibre  Stress 
12  500  Pounds 

Uniform 

Center 

Section, 

'Outside  of 
Web. 

Pound  Increase 
in  Weight. 

oer  So^.  Inch 
for  Buildings. 

:  per  Sq.  Inch, 
for  Bridges. 

load. 

Load. 

Number. 

X 

~Inch7~ 

f 

Inches. 

F 

F' 

N 

N' 

.44 

.098 

1163O 

9090  .OOO4743  .OOO7589J  C  5 

.44 

13140 

1O27O  .OO04199  .OO06718 

.46 

14710 

11490'  .0003751!  .0006001 

M 

.46 

.074 

20230 

15800  .0002046 

.0003273  C  9 

.46 

22270 

174OO  .0001858 

.0002973 

.46 

24360 

19030  .OOO  1698 

.0002717   " 

.49 

.059 

31640 

24720 

.0001046 

.0001674  C13 

.48 

37860 

29570  .OOOO875 

.0001399:   " 

.51 

44390 

34680  .0000746 

.0001  193{   " 

.52 

.049 

4621O 

36100  .O000597 

.OOOO8551  C17 

.50 

53750 

42000  .O000513  .OOO0821!   " 

.52 

61600 

48120 

.0000448  .O000717   ll 

.55 

69440 

5425O 

.O000397  .OO00636!   " 

.55 

.042 

64270 

50210 

.0000368  .O000588 

C21 

.53 

73650 

57540 

.0000321!  .OOOO7  14 

.53 

82740'   64690 

.0000286,  .0000457 

" 

.55 

91950 

71840 

.0000257  .OOO0411 

if 

.58 

101100 

78990 

.0000234 

.O000374 

« 

.58 

.037 

86140 

67300 

.000024O 

.0000384 

C25 

.56 

95990 

75000 

.0000216 

.0000345 

.56 

106450 

8317O 

.0000194 

.0000311 

u 

.57 

116910 

91340 

.0000177 

.0000283 

<* 

.59 

12737O 

9951O 

.OOOO1  62 

.O000260 

" 

.61 
.59 

.033 

11217O 
120540 

8763O 
941  7O 

.0000164!  .OO00262 
.0000153  .OOOO244 

C29 

H 

.58 

14407O 

112550 

.0000128  .O0002O4   " 

.62 

167590 

13093O 

.00001  1O1  .0000176 

" 

.64 

.029 

14268O 

111470 

.0000116  .0000186  C33 

.61 

167940 

131210 

.O0000991.  .0000158 

.62 

194090 

151630 

.00000851  .O000136 

.65 

22023O 

172060 

.OOOOO75!  .000012O   " 

.69 

246380 

19248O 

.00000671  .O000107!   " 

.70 

.025 

22775O 

177930 

.0000061 

.00.00097  C41 

.68 

256000 

20000O  .OOOOO54 

.0000086 

.68 

287370 

224510  .0000048 

.0000077   " 

.69 

318750 

249O20,  .OOOO043 

.O000069   " 

.72 

35O120 

273530  .0000039 

.0000063   " 

.79 

.020 

444520 

347280|  .0000025  .0000040  C53 

.79 

455030 

355500  .0000024 

.0000039    ' 

.78 

494250 

38613O  .OOOOO22 

.0000036! 

.79 

53347O 

416770  .0000021 

.O000033    * 

.80 

672680 

44741O  .OOOOO19  .OO00031    * 

.82 

611900 

478050  .0000018;  .0000029 

116 


SECTIONS  AND  SOLIDS 


Properties  of  Cambria  Standard  Angles. 


1 

2 

3 

4 

5 

6 

7 

8 

Area 

Distance  of 
Center  of 

Moment 

Section 

Section 

Dimensions. 

Thickness. 

Weight 
per  Foot. 

of  Section. 

Gravity  from 
Back  of 

of  Inertia 
Axis  1-1. 

Modulus 
Aus  1-1. 

Number. 

Leg. 

a  x  a 

t 

A 

x 

I 

S 

Inches. 

Inch. 

Pounds. 

Sq.  Ins. 

Inch. 

Inch«s.< 

Inchest 

A   6 

s/x   y 

Y* 

.6 

.18 

.23 

.009 

.017 

44 

it 

•fc 

.9 

.25 

.25 

.012 

.024 

Au7 

1         X    1 

% 

.8 

.24 

.30 

.022 

.031 

tt 

JL 

1.2 

.34 

.32 

.030 

.044 

tt 

44 

A 

1.5 

.44 

.34 

.037 

.056 

A   0 

1%  X   IV 

y/e 

1.1 

.30 

.36 

.044 

.049 

tt 

t 

i  ' 

1.5 
2.0 

.44 
.57 

.38 
.40 

.061 
.077 

.071 
.091 

44 

4 

2.4 

.69 

.42 

.090 

.109 

All 

IVx  IV 

Y» 

1.3 

.36 

.42 

.08 

.072 

2    4           2 

£ 

1.8 

.53 

.44 

.11 

.104 

ti 

t 

k 

2.4 

.69 

.47 

.14 

.134 

ti 

i 

ft 

2.9 

.84 

.49 

.16 

.162 

ti 

11 

% 

3.4 

.99 

.51 

.19 

.188 

it 

44 

-i* 

3.9 

1.13 

.53 

.21 

.214 

A13 

jV  x  \y 

i°s 

2.2 

.63 

.51 

.18 

.14 

tt     ' 

A 

2.8 

.82 

.53 

.23 

.19 

tt 

tt 

l*« 

3.4 

1.00 

.55 

.27 

.23 

tt 

it 

S 

4.0 

1.18 

.57 

.31 

.26 

ti 

tt 

A 

4.6 

1.34 

.59 

.35 

.30 

44 

it 

A 

5.1 

1.50 

.61 

.38 

.33 

A15 

2     x-2 

A 

2.5 

.72 

.57 

.27 

.19 

it 

M 

3.2 

.94 

.59 

.35 

.25 

ti 

it 

4.O 

1.16 

.61 

.42 

.30 

ti 

ii 

% 

4.7 

1.36 

.64 

.48 

.35 

ii 

.  ti 

7 

5.3 

1.56 

.66 

.54 

.40 

44 

ti 

A 

6.0 

1.75 

.68 

.59 

.45 

A17 

2%  x  2V. 

fa 

3.1 

.91 

.69 

.55 

.30 

it 

ii 

y 

4.1 

1.19 

.72 

.70 

.39 

tt 

•i 

$ 

5.O 

1.47 

.74 

.85 

.48 

" 

11 

% 

5.9 

1.74 

.76 

.98 

.57 

tt 

" 

l 

6.8 

7.7 

2.00 
2.25 

.78 
.81 

1.11 
1.23 

.65 
.72 

tt 

11 

TB 

8.5 

2.50 

.83 

1.34 

.80 

A19 

3     x  3 

I/ 

4.9 

1.44 

.84 

1.24 

.58 

JL 

6.1 

1.78 

.87 

1.51 

.71 

f 

ii 

/* 

7.2 

2.11 

.89 

1.76 

.83 

t« 

i 

J 

8.3 

2.44 

.91 

1.99 

.95 

it 

i 

y& 

9.4 

2.75 

.93 

2.22 

1.07 

tt 

i 

3*8 

10.4 

3.06 

.95 

2.43 

1.19 

ti 

i 

6^ 

11.5 

3.36 

.98 

2.62 

1.30 

" 

' 

ii 

12.5 

3.66 

l.OO 

2,81 

1.40 

SECTIONS  AND   SOLIDS 


117 


Properties  of  Cambria  Standard  Angles. 


9 

1O 

11 

12 

13 

1 

Radius  of 
Gyration 
Axis  1-1. 

Distance  of 
Center  of 
Gravity  from 
External  Apex 

Least  Moment  of 
Inertia 
Axis  2-2. 

Section  Modulus 
Axis  2-2. 

Least  Radius  of 
Gyration 

Axis  2-2. 

Section 
Number. 

r 

x" 

I" 

S" 

r" 

Inch. 

Inch. 

Inches.* 

Inches.3 

Inch. 

.22 

.33 

.004 

.011 

.14 

A   6 

.22 

.36 

.005 

.014 

.14 

14 

.30 

.42 

.009 

.021 

.19 

A  7 

.SO 

.45 

.013 

.028 

.19 

<t 

.29 

.48 

.016 

.034 

.19 

ii 

.38 

.51 

.018 

.035 

.24 

A   9 

.38 

.54 

.025 

.047 

.24 

H 

.37 

.57 

.033 

.057 

.24 

it 

.36 

.60 

.040 

.066 

.24 

M 

.47 

.60 

.03  1 

.053 

.30 

All 

.46 

.63 

.045 

.29 

if 

.45 

.66 

.058 

088 

.29 

it 

.44 

.69 

.070 

!ioi 

.29 

it 

.44 

.72 

.082 

.114 

.29 

it 

.43 

.75 

.094 

.126 

.29 

M 

.54 

.72 

.073 

.10 

.34 

A13 

.53 

.75 

.094 

.13 

.34 

.52 

.78 

.113 

.15 

.34 

t< 

.51 

.81 

.133 

.16 

.34 

ti 

.51 

.84 

.152 

.18 

.34 

it 

.50 

.87 

.171 

.20 

.34 

«* 

.62 

.80 

.11 

.14 

.39 

A16 

.61 

.84 

.14 

.17 

.39 

«i 

.60 

.87 

.17 

.20 

.39 

tt 

.59 

.90 

.20 

.22 

.39 

ti 

.59 

.93 

.23 

.25 

.38 

ti 

.58 

.96 

.26 

.27 

.38 

tt 

.78 

.98 

.22 

.22 

.49 

A17 

i 

1.01 
1.05 

.29 
.36 

.28 
.33 

.49 
.49 

tt 

.75 

1.08 

.41 

.38 

.48 

it 

.75 

1.11 

.46 

.42 

.48 

ti 

.74 

1.14 

.52 

.46 

.48 

tt 

.73 

1.17 

.58 

.49 

.48 

tt 

.93 

1.19 

.50 

.42 

.59 

A19 

.92 

1.22 

.61 

.50 

.59 

.91 

1.26 

.72 

.57 

.58 

tt 

.91 

1.29 

.82 

.64 

.58 

tt 

.90 

1.32 

.92 

.70 

.58 

ti 

.89 

1.35 

1.02 

.76 

.68 

it 

.88 

1.38 

1.12 

.81 

.58 

it 

.88 

1.41 

1.22 

.86 

.58 

M 

[IS 


SECTIONS  AND   SOLIDS 


Properties  of  Cambria  Standard  Angles. 


1 

2 

3 

4 

5 

6 

7 

8 

Section 
Number. 

Dimensions. 

Thickness 

Veight 
per  Foot. 

Irea 
of  Section. 

Distance  of 
Center  of 
Gravity  from 
Back  of 
Leg. 

Moment 
of  Inertia 
iiis  1-1. 

Section 
Modulus 
Iiis  1-1. 

a  la 

t 

A 

X 

I 

S 

Inches. 

Inch. 

Pounds. 

Sq.  Ins. 

Inches. 

Inchest 

Inches.' 

A21 

3%x  3^ 

fr 

7.2 

2.09 

.99 

2.45 

.98 

i 

•*  it 

y* 

8.5 

2.49 

1.01 

2.87 

1.15 

4 

ti 

fr 

0.8 

2.88 

1.04 

3.26 

1.32 

1 

it 

| 

11.1 

3.25 

1.O6 

3.64 

1.49 

I 

i* 

124 

3.63 

1.08 

3.99 

1.65 

( 

14 

/8 

13.6 

3.99 

1.1O 

4.33 

1.81 

t 

14 

11 

14.8 

4.34 

1.12 

4.65 

1.96 

t 

il 

%. 

16.0 

4.69 

1.15 

4.96 

2.11 

t 

ti 

ll 

17.1 

5.O3 

1.17 

5.25 

2.25 

t 

it 

*  7 

/a 

18.3 

5.36 

1.19 

5.53 

2.39 

A23 

4x4 

T8* 

8.2 

2.41 

1.12 

3.71 

1.29 

44 

tt 

B 

9.8 

2.86 

1.14 

4.36 

1.52 

it 

44 

I73 

11.3 

3.31 

1.16 

4.97 

1.75 

it 

it 

8 

12.8 

3.75 

1.18 

5.56 

1.97 

tt 

ti 

r5 

14.3 

4.19 

1.21 

6.12 

2.19 

tt 

tt 

% 

15.7 

4.62 

1.23 

6.66 

2.40 

it 

it 

u 

17.1 

6.03 

1.25 

7.17 

2.61 

it 

4t 

^ 

18.5 

5.44 

1.27 

7.66 

2.81 

u 

44 

19.9 

5.84 

1.29 

8.14 

3.01 

14 

it 

B 

/'a 

21.2 

6.24 

1.31 

8.59 

3.20 

A27 

6x6 

% 

14.9 

4.36 

1.64 

15.39 

3.53 

ti 

tt 

X 

17.2 

5.O6 

1.66 

17.68 

4.07 

u 

it 

M 

19.8 

5.75 

1.68 

19.91 

4.61 

u 

t 

$ 

21.9 

6.44 

1.71 

22.07 

5.14 

u 

t 

ox£ 

24.2 

7.11 

1.73 

24.16 

6.86 

11 

t 

li 

26.5 

7.78 

1.75 

26.19 

6.17 

tt 

4 

74. 

28.7 

8.44 

1.78 

28.15 

6.66 

tl 

t 

1| 

31.O 

9.09 

1.80 

30.06 

7.15 

it 

4 

y  H 

33.1 

9.74 

1.82 

31.92 

7.63 

ti 

4 

ij 

35.3 

10.38 

1.84 

33.72 

8.11 

il 

it 

I 

37.4 

11.0O 

1.86 

35.46 

8.67 

A35 

8      x  8 

K 

26.4 

7.75 

2.19 

48.65 

8.37 

tt 

ft 

29.6 

8.69 

2.21 

54.09 

9.34 

tt 

i 

F  / 

/8 

32.7 

9.61 

2.23 

59.43 

10.3O 

tt 

4 

u 

35.8 

10.53 

2.25 

64.64 

11.25 

i 

i 

% 

38.9 

11.44 

2.28 

69.74 

12.18 

i 

» 

$ 

42.O 

12.34 

2.30 

74.72 

13.11 

t 

t 

7X 

''Q 

45.O 

13.24 

2.32 

79.58 

14.02 

i 

4 

if 

48.1 

14.13 

2.34 

84.34 

14.91 

i 

i 

1 

51.0 

15.00 

2.37 

88.98 

15.8O 

« 

4 

1^ 

54.0 

15.88 

2.39 

93.53 

16.67 

tt 

t 

i5 

56.9 

16.74 

2.41 

97.97 

17.53 

SECTIONS  AND   SOLIDS 


119 


Properties  of  Cambria  Standard  Angles. 


9 

1O 

11 

12 

13 

1 

Radios  of 

Distance  of 
Center  of 

Least  Moment  of 

Section  Modulus 

Least  Radius  of 

Gyration 
Axis  1-1. 

Gravity  from 
External  Apex. 

Inertia 
Aiis2-2. 

Axis  2-2. 

Gyratioa 
Axis  2-2. 

Section 

Nomfer. 

r 

x" 

I" 

8" 

r" 

Inches. 

Inches. 

Inches.  « 

Inches.' 

Inch. 

1.08 

1.4O 

.99 

.71 

.69 

A21 

1.07 

1.43 

1.16 

.81 

.68 

1.07 

1.46 

1.33 

.91 

.68 

1.O6 

.50 

1.50 

l.OO 

.68 

1.O5 

:  .53 

1.66 

1.09 

.68 

1.04 

.56 

1.82 

1.17 

.68 

1.04 

.59 

1.97 

1.24 

.67 

1.03 

.62 

2.13 

1.31 

.67 

1.O2 

.65 

2.28 

1.38 

67 

1.O2 

.68 

2.43 

1.45 

.67 

1.24 

.58 

1.50 

.95 

.79 

A23 

1.23 

.61 

1.77 

1.1O 

.79 

1.23 

.64 

2.02 

1.23 

.78 

1.22 

.67 

2.28 

1.36 

.78 

1.21 

1.71 

2.52 

1.48 

.78 

1.20 

1.74 

2.76 

1.59 

.77 

1.19 

1.77 

3.0O 

1.7O 

.77 

1.19 

1.8O 

3.23 

1.80 

.77 

1.18 

1.83 

3.46 

1.89 

.77 

1.17 

1.86 

3.69 

1.99 

.77 

1.88 

2.32 

6.19 

2.67 

1.19 

A27 

1.87 

2.34 

7.13 

3.04 

1.19 

1.86 

2.38 

8.O4 

3.37 

.18 

1.85 

2.41 

8.94 

3.70 

.18 

1.84 

2.45 

9.81 

4.01 

.17 

1.83 

2.48 

10.67 

4.31 

.17 

1.83 

2.51 

11.52 

4.59 

.17 

1.82 

2.54 

12.35 

4.86 

.17 

1.81 

2.57 

13.17 

6.12 

.16 

1.80 

2.6O 

13.98 

6.37 

.16 

1.8O 

2.64 

14.78 

6.61 

.16 

2.61 

3.09 

19.56 

6.33 

.59 

A35 

2.5O 

3.12 

21.79 

6.98 

.58 

2.49 

3.16 

23.97 

7.60 

.58 

2.48 

3.19 

26.13 

8.2O 

.68 

2.47 

3.22 

28.24 

8.77 

.67 

2.46 

3.25 

30.33 

9.33 

.57 

2.45 

3.28 

32.38 

9.86 

.56 

2.44 

3.32 

34.4O 

10.38 

.56 

2.44 

3.35 

36.40 

1O.88 

.56 

2.43 

3.38 

38.38 

11.36 

.56 

2.42 

3.41 

40.33 

11.83 

.56 

120 


SECTIONS  ANIfcSOLIDS 


Properties  of  Cambria  Standard  Angles. 


1 

2 

3 

4 

5 

6 

7 

8 

Section 

Dimensions. 

Thickness 

Weight 
per  Foot 

irea 
of  Section, 

Distance  of  Center 
of  Gravity  from 
Back  of  Longer 
Leg. 

Moment  o 
Inertia 
Aiis  1-1. 

Section 
Modulus 
Aiis  1-1. 

Number. 

b  i  a 

t 

A 

X 

r 

S 

Inches. 

Inch. 

Pounds. 

Sq.  Ins. 

Inch. 

Inches.* 

Inches.3 

A91 

2^*2 

138 

2.8 

.81 

.61 

.29 

.20 

rm  tt 

B 

3.7 

1.07 

.54 

.37 

.25 

" 

" 

ft 

4.5 

1.31 

.56 

.45 

.31 

" 

(i 

B 

5.3 

1.55 

.58 

.51 

.36 

ii 

»l 

ft 

6.1 

1.78 

.60 

.58 

.41 

ii 

II 

8 

6.8 

2.00 

.63 

.64 

.46 

44 

M 

X 

7.6 

2.22 

.65 

.69 

.51 

A93 

3     x2K 

K 

4.5 

1.32 

.66 

.74 

.40 

it 

(i 

A 

5.6 

1.63 

.68 

.90 

.49 

ti 

i 

S 

6.6 

1.93 

.71 

1.04 

.58 

" 

i 

]MJ 

7.6 

2.22 

.73 

1.18 

.66 

II 

i 

8.5 

2.50 

.75 

1.3O 

.74 

ii 

* 

•fe 

9.5 

2.78 

.77 

1.42 

.82 

U 

4 

10.4 

3.05 

.79 

1.53 

.90 

A95 

3^x2% 

\i 

4.9 

1.44 

.61 

.78 

.41 

tt 

A 

6.1 

1.78 

.64 

.94 

.50 

M 

ii 

% 

7.2 

2.11 

.66 

1.09 

.59 

II 

it 

8.3 

2.44 

.68 

1.23 

.68 

" 

" 

B 

9.4 

2.75 

.70 

1.36 

.76 

M 

*i 

» 

10.4 

3.06 

.73 

1.49 

.84 

" 

M 

II 

11.5 

3.36 

.75 

1.61 

.92 

it 

M 

H 

12.5 

3.66 

.77 

1.72 

.99 

" 

If 

K 

13.4 

3.94 

.70 

1.83 

1.07 

A97 

3X*3 

A 

6.6 

1.94 

.81 

1.58 

.72 

•*  tt 

51 

7.9 

2.3O 

.83 

1.85 

.85 

.•* 

ii 

g 

9.1 

2.66 

.85 

2.09 

.98 

«' 

M 

10.2 

3.00 

.88 

2.33 

1.10 

M 

ii 

I95 

11.4 

3.34 

.90 

2.55 

1.21 

M 

1 

S/ 

12.5 

3.68 

.92 

2.76 

1.33 

41 

• 

H 

13.6 

4.00 

.94 

2.96 

1.44 

II 

1 

II 

14.7 

4.32 

.96 

3.15 

1.54 

« 

1 

II 

15.8 
16.8 

4.63 
4.93 

.98 
1.00 

3.33 
3.50 

1.65 
1.75 

A99 

4x3 

ft 

7.2 

2.09 

.76 

1.65 

.73 

M 

'  " 

78 

8.5 

2.49 

.78 

1.92 

.87 

tt 

ti 

J_ 

9.8 

2.88 

.80 

2.18 

.99 

tt 

44 

12 

11.1 

3.25 

.83 

2.42 

1.12 

14 

" 

1*8 

12.4 

3.63 

.85 

2.66 

1.23 

t 

ii 

% 

13.6 

3.99 

.87 

2.87 

1.35 

c 

44 

11 

14.8 

4.34 

.89 

3.08 

1.46 

• 

ii 

/4 

16.0 

4.69 

.92 

3.28 

1.57 

t 

ti 
ti 

II 

17.1 
18.3   1 

5.03 
5.36 

.94 
.96 

3.47 
3.66 

1.68 
1.79 

SECTIONS   AND   SOLIDS 


121 


Properties  of  Cambria  Standard  Angles 


9                10               11 

12 

13 

14 

15 

1 

Radius  of 
Gyration 
Axis  1-1. 

Distance  of  Center 
of  Gravity  from 
Back  of  Shorter 
I*. 

Moment  of 
Inertia 
Aiis2-2. 

Section 
Modulus 
Aiis  2-2. 

Radius  of 
Gyration 
Axis  2-2. 

Tangent 
of  Angle 

Least  Radius 
of  Gyration 
Axis  3-3. 

Section 

r 

x'                 I' 

S' 

r> 

a 

•      r// 

Number. 

Inch. 

Inch. 

Inches.* 

Inches.3 

Inches. 

Inch. 

.60 

.76 

.51 

.29 

.79 

.632 

.43 

A91 

.59 

.79 

.65 

.38 

.78 

.626 

.42 

.58 

.81 

.79 

.47 

.78 

.620 

.42 

.58 

.83 

.91 

.55 

.77 

.614 

.42 

.57 

.85 

1.03 

.62 

.76 

.607 

.42 

.56 

.88 

1.14 

.70 

.75 

.600 

.42 

.56 

.90 

1.24 

.77 

.75 

.592 

.42 

.75 

.91 

.1.17 

.56 

.95 

.684 

.53 

A93 

.74 

.93 

1.42 

.69 

.94 

.680 

.53 

.74 

.96 

1.66 

.81 

.93 

.676 

.52 

.73 

.98 

1.88 

.93 

.92 

.672 

.52 

.72 

1.00 

2.O8 

1.O4 

.91 

.666 

.52 

.72 

1.O2 

2.28 

1.15 

.91 

.661 

.52 

.71 

1.04 

2.46 

1.26 

.90 

.655 

.52 

.74 

1.11 

1.80 

.75 

1.12 

.506 

.54 

A95 

.73 

1.14 

2.19 

.93 

1.11 

.501 

.54 

.72 

1.16 

2.56 

1.09 

1.1O 

.496 

.54 

.71 

1.18 

2.91 

1.26 

1.O9 

.491 

.54 

.70 

1.2O 

3.24 

1.41 

1.09 

.486 

.53 

.70 

1.23 

3.55 

1.56 

1.O8 

.480 

.53 

.69 

1.25 

3.85 

1.71 

1.07 

.472 

.53 

.69 

1.27 

4.13 

1.85 

1.O6 

.468 

.53 

i 

.68 

1.29 

4.40 

1.99 

1.O6 

.461 

.54 

• 

.90 

1.O6 

2.33 

.95 

1.10 

.724 

.63 

A97 

.90 

1.08 

2.72 

1.13 

1.O9 

.721 

.62 

.89 

1.10 

3.1O 

1.29 

1.O8 

.718 

.62 

.88 

1.13 

3.45 

1.45 

1.07 

.714 

.62 

.87 

1.15 

3.79 

1.61 

1.07 

.711           .62 

.87 

1.17 

4.11 

1.76 

1.O6 

.707           .62 

.86 

1.19 

4.41 

1.91 

1.05 

.703 

.62 

.85 

1.21 

4.7O 

2.05 

1.04 

.698 

.62 

.85 

1.23 

4.98 

2.20 

1.O4 

.694 

.62 

.84 

1.25 

5.24 

2.33 

1.03 

.689 

.63 

.89 

1.26 

3.38 

1.23 

1.27 

.554 

.65 

A99 

.88 

1.28 

3.96 

1.46 

1.26 

.551 

.64 

.87 

1.30 

4.52 

1.68 

1.25 

.547 

.64 

.86 

1.33 

5.05 

1.89 

1.25 

.543 

.64 

.86 

1.35 

5.55 

2.O9 

1.24 

.538 

.64 

.86 

1.37 

6.03 

2.3O 

1.23 

.534 

.64 

.84 

1.39 

6.49 

2.49 

1422 

.529 

.64 

.84 

1.42 

6.93 

2.68 

1.22 

.524 

.64 

.83 

1.44 

7.35 

2.87 

1.21 

.518 

.64 

.83 

1.46 

7.75 

3.05 

1.20 

.512 

.64 

122 


SECTIONS  AND   SOLIDS 


Properties  of  Cambria  Standard  Angles. 


1 

2 

3 

4 

6 

6 

7 

8 

Section 
Nuznbcr. 

Dimensions. 

Thickness. 

Weight 
per  Foot. 

Area  of 

Section. 

Distance  of  Center 
of  Gravity  from 
Back  of  Longer 
Leg. 

Moment  of 
Inertia 
Axis  1-1. 

Section 
Modulus 
Axis  1-1. 

b  x  a 

t 

A 

x. 

I 

S 

Inches. 

Inch. 

Pounds. 

Sq.  Ins. 

Inch. 

Inches.* 

Inches.3 

A101 

5     x3 

A 

8.2 

2.41 

.68 

1.75 

.75 

*/ 

9.8 

2.86 

.70 

2.04 

.89 

44 

44 

Jr 

11.3 

3.31 

.73 

2.32 

1.02 

44 

44 

I/ 

12.8 

3.75 

.75 

2.58 

1.15 

44 

41 

I99 

14.3 

4.19 

.77 

2.83 

1.27 

44 

14 

5/ 

15.7 

4.61 

.80 

3.06 

1.39 

" 

4t 

i^ 

17.1 

5.03 

.82 

3.29 

1.61 

ti 

M 

*A 

18.5 

5.44 

.84 

3.51 

1.62 

tt 

44 

M 

19.9 

5.84 

-.86 

3.71 

1.74 

M 

" 

/S 

21.2 

6.24 

.88 

3.91 

1.85 

A103 

5     x3V£ 

A 

8.7 

2.56 

.84 

2.72 

1.02 

$8 

10.4 

3.05 

.86 

3.18 

1.21 

44 

44 

A 

12.O 

3.53 

.88 

3.63 

1.39 

44 

44 

8 

13.6 

4.00 

.91 

4.05 

1.56 

44 

14 

15.2 

4.47 

.93 

4.45 

1.73 

14 

44 

s 

16.8 

4.93 

.95 

4.83 

1.90 

4» 

44 

i4 

18.3 

5.38 

.97 

5.20 

2.06 

44 

44 

•% 

19.8 

5.82 

1.00 

5.55 

2.22 

44 

44 

9 

21.3 

6.25 

1.02 

5.89 

2.37 

44 

44 

/ 

22.7 

6.68 

1,04 

6.21 

2.52 

44 

*• 

H 

24.2 

7.09 

1.06 

6.52 

2.67 

A  105 

6     x  Q% 

Ys 

11.7 

3.43 

.79 

3.34 

1.23 

13.5 

3.97 

.81 

3.81 

1.41 

41 

41 

15.3 

4.50 

.83 

4.25 

1.59 

44 

44 

» 

17.1 

5.03 

.86 

4-67 

1.77 

44 

tt 

5X 

18.9 

5.55 

.88 

5.08 

1.94 

44 

44 

H 

20.6 

6.06 

.90 

5.47 

2.11 

44 

t 

I? 

22.4 

6.57 

.93 

5.84 

2.27 

44 

I 

13 

24.0 

7.06 

.95 

6.20 

2.43 

44 

4 

/a 

25.7 

7.55 

.97 

6.55 

2.59 

44 

4 

u 

27.3 

8.03 

.99 

6.88 

2.74 

" 

t 

1 

28.9 

8.5O 

1.01 

7.21 

2.90 

A107 

6     x4 

3/ 

12.3 

3.61 

.94 

4.90 

1.60 

JC 

14.3 

4.19 

.96 

5.60 

1.85 

44 

M 

I" 

16.2 

4.75 

.99 

6.27 

2.08 

44 

44 

'9 

18.1 

5.31 

1.01 

6.91 

2.31 

41 

44 

c/ 

2O.O 

5.86 

1.03 

7.52 

2.54 

44 

tt 

if 

21.8 

6.41 

1.06 

8.11 

2.76 

If 

14 

k 

23.6 

6.94 

1.O8 

8.68 

2.97 

M 

44 

ii 

25.4 

7.47 

1.10 

9.23 

3.18 

M 

4» 
«i 

" 

1 

27.2 
28.9 
30.6 

7.99 
8.5O 
9.0O 

1.12 
1.14 
1.17 

9.75 
10.26 
10.75 

3.39 
3.69 
8.79 

SECTIONS   AND   SOLIDS 


123 


Properties  of  Cambria  Standard  Angles. 


9 

1O 

11 

12 

13 

14 

15 

1 

Radius  of 
Gyration 
Axis  1-1. 

Distance  of  Center 
of  Gravity  from 
Back  of  Shorter 
Leg. 

Moment  of 
Inertia 
Aiis  2-2. 

Section 
Modulus 
Aiis  2-2. 

Radius  of 
Gyration 
Axi?  2-2. 

Tangent 
of  Angle 

Least  Radius 
of  Gyration 
Aiis  3-3. 

Section 

Ki     Vi 

r 

x' 

I' 

S' 

r' 

a 

r" 

'  • 

Inch. 

Inches. 

Inches.4 

Inches/' 

Inch. 

Inch. 

.85 

1.68 

6.26 

1.89 

1.61 

.368 

.66 

Alpl 

.84 

1.70 

7.37 

2.24 

1.61 

.364 

.65 

.84 

1.73 

8.43 

2.58 

1.60 

.361 

.65 

i« 

.83 

1.75 

9.45 

2.91 

1.59 

.357 

.65 

k» 

.82 

1.77 

10.43 

3.23 

1.58 

.353 

.65 

H 

.82 

1.80 

11.37 

3.55 

1.57 

.349 

.64 

U 

.81 

1.82 

12.28 

3.86 

1.56 

.345 

.64 

If 

.80 

1.84 

13.15 

4.16 

1.55 

.340 

.64 

U 

.80 

1.86 

13.98 

4.46 

1.55 

.336 

.64 

it 

.79 

1.88 

14.78 

4.75 

1.54 

.331 

.64 

(« 

1.03 
1.02 

1.59 
1.61 

6.60 

7.78 

1.94 
2.29 

1.61 
1.60 

.489 
.485 

g 

A  103 

1.01 

1.63 

8.90 

2.64 

1.59 

.482 

.76 

1C 

1.01 

1.66 

9.99 

2.99 

1.58 

.479 

•75 

ti 

1.00 

1.68 

11.  03 

3.32 

1.57 

.476 

.75 

'« 

.99 

1.70 

12.03 

3.65 

1.66 

.472 

.75 

" 

.98 

1.72 

12.99 

3.97 

1.5ft 

.468 

.75 

«« 

.98 

1.75 

13.92 

4.28 

1.55 

.464 

.75 

1 

.97 

1.77 

14.81 

4.58 

.54 

.460 

.75 

* 

.96 

1.79 

15.67 

4.88 

.53 

.455 

.75 

c 

.96 

1.81 

16.49 

5.17 

.53 

,451 

.75 

f 

.99 

£.04 

12.86 

3.24 

.94 

.350 

.77 

A105 

.98 

2.06 

14.76 

3.75 

.93 

.347 

.76 

4 

.97 

208 

16.59 

4.24 

.92 

.344 

.76 

t 

.96 

2.11 

18.37 

4.72 

.91 

.341 

.75 

« 

.96 

2.13 

2O.08 

5.19 

.90 

.338 

.75 

1 

.95 

2.15 

21.74 

5.65 

.89 

.334 

.75 

t 

.94 

2.18 

23.34 

6.10 

.89 

.331 

.75 

* 

.94 

2.20 

24.89 

6.55 

.88 

.327 

.75 

* 

.93 

2.22 

26.39 

6.98 

.87 

.323 

.75 

1 

.93 

2.24 

27.84 

7.41 

.86 

.320 

.75 

1 

.92 

2.26 

29.15 

7.8O 

1.85 

.317 

.75 

I 

1.17 

1.94 

13.47 

3.32 

1.93 

.446 

.88 

A1O7 

1.16 

1.96 

15.46 

3.83 

1.92 

.443 

.87 

1.15 

1.99 

17.40 

4.33 

1.91 

.440 

.87 

<» 

1.14 

2.01 

19.26 

4.83 

1.9O 

.438 

.87 

M 

1.13 

2.03 

21.07 

5.31 

1.9O 

.434 

.86 

1 

1.13 

2.O6 

22.82 

5.78 

1.89 

.431 

.86 

* 

1.12 

2.08 

24.51 

6.25 

1.88 

.428 

.86 

« 

1.11 

2.10 

26.15 

6.7O 

1.87 

.425 

.86 

» 

1.11 

2.12 

27.73 

7.16 

1.86 

.421 

.86 

' 

1.10 

2.14 

29.26 

7.59 

1.86 

.418 

.86 

it 

1.09 

2.17 

30.75 

6.02 

1.85 

.414 

86 

" 

124 


SECTIONS  AND   SOLIDS 

Properties  of  Cambria  T-Bars. 


EQUJLZ.   X-EGS. 


1 

2-|3|4            6 

6 

7 

8 

9 

Dimensions. 

Section 
Humber. 

Width 
of 

Flange. 

Depth 
of 
Bar. 

Thickness 
of 
Flange. 

Thickness 
of 
Stem. 

Weight 
per 
Foot. 

ire* 
of 
Section. 

Distance  of 
Center  of 
Gravity 
from  Outside 
of  Flange. 

Moment  of 
Inertia 
Axis  1-1. 

b 

1 

8  to  n' 

t  to  t' 

A 

•x. 

I 

Inches. 

Inches. 

Inch. 

Inch. 

Pounds. 

Sq.  Ins. 

Inch. 

Inches.* 

T      5 

1 

1 

K*»* 

K«oA 

1.0 

.27 

.29 

.02 

T181 

1% 

1% 

A  '  A 

A  "  A 

1.4 

.41 

.33 

.04 

T183 

1A 

IA 

A  "  K 

A  "  A 

1.6 

.45 

.34 

.05 

T187 

1/4 

IK 

A  "  % 

A  "  X 

1.7 

.48 

.36 

.06 

T189 

1% 

1% 

A  "  /4 

A  "  /4 

1.9 

.55 

.39 

.08 

T   37 
T   39 

2 

2 

2 
2 

&  •'  $ 

^"S 

3.7 

4.4 

1.07 
1.28 

.59 
.61 

.37 
.43 

T    41 

2*4 

2K 

/4  "  A 

54  "  A 

4.2 

1.21 

.68 

.51 

T    42 

2^-1 

A   "  78 

6.0 

1.46 

.67 

.64 

T   49 

2/<2 

2% 

A  "  % 

A  "  % 

5.6 

1.63 

.73 

.87 

T    67 

3 

3 

A  "  %  A  "  % 

6.8 

1.99 

.86 

1.58 

T   69 

3 

3 

%  "  A  I  /*  "  A 

7.9 

2.31 

.88 

1.82 

T   73 

3 

3 

/1»  "  A 

10.1 

2.96 

.93 

2.27 

T  97 

Q% 

3% 

%  "  A 

3J"A 

9.3 

2.74 

.99 

3.10 

T108 

4 

4 

%"  A 

3/^  "A 

10.9 

3.19 

1.12 

4.54 

UNEQUAL  LEGS. 

T185 

13* 

1A 

AtoK 

A  to  A 

1.5 

.44 

.29 

.04 

T   22 

2K 

1/4 

A  "  A 

A"  A 

3.0 

.86 

.30 

.08 

T   27 

2% 

1/4 

4.5 

1.31 

.43 

.21 

T    56 

2>2 

3 

%  "A  %"£ 

7.2 

2.10 

.95 

1.72 

T   62 

2% 

2 

7.5 

2.21 

.74 

.83 

T   65 

3 

2% 

%  "  A  !  %  to  A 

7.2 

2.11 

.71 

1.08 

T   84 

3 

4 

%  "  A  !  Ys  "  A 

9.3 

2.74 

1.27 

4.12 

T101 

3% 

4 

%  "  A   %  "  A 

10.0 

2.94 

1.20 

4.33 

T120 

4% 

2% 

A  "  A 

A  "% 

8.0 

2.29 

.57 

1.04 

T138 

4% 

3^2 

7      II    • 

is     IB 

II 

14.9 

4.37 

1.09 

4.89 

T140 

4% 

3/4 

A  "A 

15.9 

4.65  i     1.11 

5.08 

T169 

5 

3 

13.6 

3.99  ;       .72 

2.42 

SECTIONS   AND   SOLIDS 


125 


Properties  of  Cambria  T-Bars 


EQTCJAJL.  LEGS. 


10       ir 

12     |     13 

14 

15        I        16        i         1 

Coef.  of  Strength. 

Section 

Radius  of 

Moment  of 

Section 

Radiusof 

Modulus 
iiis  1-1. 

Gyration 
iiis  1-1 

Inertia 
liis2-2. 

Modulus 
liis  2-2. 

Gyration 
Aiis  2-2. 

For  Fibre  Stress 
of  16  000  Lbs. 

••a* 

For  Fibre  Stress 
of  12  500  Lbs. 
per  Square 
Inch. 

Section 
Number. 

8 

r 

1' 

S' 

r' 

F 

F' 

"incheaT 

Inch. 

Inches.* 

Inches.* 

Inch. 

.03 

.30 

.01 

.02 

.21 

350 

27O 

T       6 

.05 

.32 

.02 

.04 

.25 

560 

44O 

T181 

.06 

.33 

.03 

.05 

.26 

630 

490 

T183 

.07 

.35 

.03 

.05 

427 

70O 

55O 

T187 

.08 

.39 

.05 

.07 

.29 

89O 

69O 

T189 

.26 

.59 

.18 

.18 

.42 

277O 

2160 

T   37 

.31 

.59 

.23 

.23 

.42 

3330 

26OO 

T   39 

.32 

.66 

.24 

.21 

.45 

3440 

2690 

T   41 

.40 

.66 

.32 

.29 

.47 

43OO 

336O 

T   42 

.49 

.74 

.44 

.35 

.52 

525O 

410O 

T   49 

.74 

.90 

.75 

.50 

.62 

78  6O 

6140 

T   67 

.86 

.90 

.92 

.61 

.64 

918O 

718O 

T    69 

1.10 

.88 

1.2O 

.80 

.64 

1171O 

915O 

T    73 

1.23 

1.O8 

1.42 

.81 

.73 

13140 

1026O 

T   97 

1.58 

1.21 

2.11 

1.06 

.83 

1685O 

13170 

T108 

TJTNTEQTJAX.  LEGS. 

.05 

.29 

.03 

.01 

.28 

5OO 

39O 

T186 

.00 

.31 

.28 

.22 

.58 

93O 

73O 

T   22 

.16 

.40 

.47 

.38 

.60 

170O 

1320 

T   27 

.84 

.91 

.53 

.42 

.50 

893O 

698O 

T   56 

.66 

.62 

.64 

.46 

.54 

7020 

5490 

T   62 

.60 

.64 

.90 

.60 

.66 

6400 

6000 

T   65 

|.*1 

1.24 

.90 

.60 

.58 

1609O 

12570 

T   84 

f.t»4 

1.23 

1.42 

.81 

•7Q 

1647O 

1286O 

T101 

.54 

.68 

2.51 

1.12 

1.O5 

576O 

45OO 

T12O 

2.03 

1.06 

3.68 

1.64 

.92 

2161O 

1688O 

T138 

2.13 

1.O5 

3.73 

1.66 

.90 

2269O 

1772O 

T140 

1.06 

.78 

5.42 

2.17 

1.17 

11340 

8860 

T169 

126 


SECTIONS  AND   SOLIDS 


Properties  of  Cambria  Z-Bars 


1 

234 

5 

6 

7     |     8 

9 

1O 

11      12 

Thick- 

Radius 

.     Radius 

Section 

Depth 
of 

Length 
of 

ness 
of 

Web 

V'ght 

Area 
of 

Moment 
of. 
Tn6rti& 

Section 
Modulus 

of 
Gyra- 
lion 

Moment 
of 
Inertia 

Section 
Mod- 
ulus 

of 
Gyra- 
tion 

Num- 

Bar. 

Legs. 

and 

per 

pXrtt 

Section. 

Axis  1-1. 

Axis  1-1. 

Axis 

Axis  2-2. 

Axis 

Axis 

ber. 

Legs. 

1  001. 

1-1. 

2-2. 

2-2. 

b 

a         t 

j^ 

I 

$ 

r 

I' 

S' 

r' 

"Inches! 

Inches.1  Inch.  !  Lbs. 

SqTlnsT 

Indies/ 

iTcheZ3 

Ins.  j  Inches/ 

IM.» 

Inch. 

Z    5 

3j 

&u 

.,x 

6.7 

1.97     2.87 

1.92 

1.21 

2.81 

1.10 

1.19 

" 

2/4 

£ 

8.4    2.48    3.64 

2.38 

1.21 

3.64 

1.40  1.21 

Z    9 

318 

|B 

34 

9.7    2.86 

3.85 

2.57 

1.16    3.92 

1.57  1.17 

tt 

H5 

11.4 

3.36 

4.57 

2.98 

1.17    4.75 

1.88.1.19 

Z13 

3* 

Jill 

I/ 

12.5 

3.69 

4.59 

3.06 

1.12    4.85 

1.99  1.15 

2x4 

^ 

14.2 

4.18 

6.26 

3.43 

1.12    5.68 

;2.30|1.17 

Z21 

4™ 

3iV 

¥ 

8.2 

2.41 

6.28 

3.14 

1.62    4.23 

1.  441  1.33 

tt 

4  A 

3% 

10.3 

3.03 

7.94 

3.91 

1.62    6.46 

1.  84  i  1.34 

tt 

4k 

3  13- 

X8 

12.4 

3.66 

9.63    4.67 

1.62     6.77 

2.26,1.36 

Z25 

4 

3  fe 

A 

13.8 

4.05 

9.66    4.83 

1.64    6.73 

2.37'l.29 

tt 

4  V 

3  •? 

? 

15.8 

4.66 

11.18 

5.50 

1.55 

7.96 

2.77  1.81 

" 

Si 

17.9 

5.27 

12.74 

6.18 

1.55 

9.263.19  1.32 

Z29 

4 

3/B 

5^ 

18.9 

5.55 

12.11 

6.05 

1.48 

8.733.18  1.25 

" 

ftlz 

|! 

2O.9 

6.14 

13.52 

6.65 

1.48 

9.95 

3.68  1.27 

M 

4/1 

3^ 

8 

23.0 

6.75 

14.97 

7.26 

1.49 

11.24  4.00  1.29 

Z37 

5 

31^ 

ft 

11.6 

3.40 

13.36 

6.34 

1.98 

6.18  2.00  1.35 

»* 

g  JL 

3-8s 

42 

13.9 

4.10 

16.18 

6.39 

1.99 

7.6ft 

2.45  1.37 

li 

5}<J 

85? 

i7i 

16.4 

4.81 

19.07 

7.44 

1.99 

9.20,2.92  1.38 

Z41 

5 

3% 

X-3 

17.9 

5.25 

19.19 

7.68 

1.91 

9.O5  3.02  1.31 

*' 

6  A 

3i6n 

i*r 

20.2 

6.94 

21.83 

8.62 

1.92  1O.51 

3.47  1.33 

ii 

5% 

S$i 

% 

22.6 

6.64  24.53 

9.57 

1.92,12.06:3.941.35 

Z45 

tt 

5j 

i1^ 

M 

23.7 

26.O 

6.96  23.68 
7.6426.16 

9.47 
10.34 

1.84  11.373.91  1.28 
1.85  12.83  4.37  1.3O 

M 

5)i 

3% 

il     28.3 

8.33 

28.7O  11.20 

1.86:14.374.841.31 

Z53 

6^ 

/'a 

15.6 

4.59  25.32 

8.4412.35!   9.11 

2.75 

1.41 

3/8 

rt 

18.3 

6.39  29.80 

9.83 

2.35  1O.94 

3.27 

1.43 

" 

1& 

Q5/ 

k 

21.O 

6.19  34.36 

11.22 

2.36,12.87 

3.81 

1.44 

Z57 

6 

39 

/T 

22.7 

6.68 

34.64  11.55 

2.28  12.69  3.91 

1.37 

% 

25.4 

7.46 

S8.87  12.82 

2.28  14.41 

4.44  1.39 

tt 

61  ^ 
78 

3% 

i! 

28.1 

8.25 

43.18  14.10  2.29.18.34 

4.98  1.41 

Z61 

6 

873 

/* 

29.3 

8.63 

42.12  14.04 

2.21  16.44 

4.94 

1.34 

ii 

& 

i& 

I 

31.9 
34.6 

9.39 
10.17 

46.13  1  5.22  i  2.22  \  17.27  5.47 
5O.22  16.40  2.22  19.18  6.02 

1.36 
1.37 

Z67 

7% 

3 

16.3 

4.78 

38.19  10.18 

2.83 

6.69 

1.99 

1.08 

Z73 

8 

3 

M 

22.1 

6.50 

66.54  14.14  2.96]    7-01 

2.65 

1.0* 

SECTIONS  AND   SOLIDS 

Properties  of  Cambria  Z-Bars 


127 


r 

t 

i 
it 

1rt 

•\  — 

- 

E 

1 

t 

13   i   14      15    |    16 

17       18 

1 

Least  Radius 

Coef.  of  Strength. 

Coef.  of  Deflection. 

Tangent 

of 

For  Fibre  Stress  Tor  Fibre  Stress 

of  Angle 

Gyration 

of  16  000 
Pounds  per 

of  12500 
Pounds  per 

Uniform  Load. 

Center  Load. 

Section 

a 

Aiis  3-3. 

Square  Inch. 

Square  Inch. 



Number. 

T" 

Inch! 

F 

F' 

N 

H' 

.986 

.55 

2040O 

1600O 

.000270 

.OO0432 

Z  5 

1.000 

.55 

25400 

1980O 

.000213 

.000341 

" 

.990 

.54 

274OO 

2140O 

.OOO2O1 

.OOO322 

Z  9 

.975 

.55 

3180O 

248OO 

.OOO17O 

.000272 

M 

.965 

.53 

3260O 

25500 

.000169 

.000271 

Z13 

.951 

.54 

36600 

286OO 

.000148 

.000236 

" 

.778 

.67 

33500 

26200 

.OOO  124 

.000198 

Z21 

.788 

.68 

41700 

3260O 

.OOOO98 

.OOO  156 

" 

.798 

.69 

49800 

3890O 

.000081 

.OOO  129 

tt 

.794 

.66 

5150O 

4O20O 

.OOO08O 

.000129 

Z25 

.804 

.67 

5870O 

4590O 

.OO0069 

.O00111 

.814 

.68 

65900 

61500 

.000061 

.OOO098 

M 

.808 

.65 

64600 

6050O 

.000064 

.000103 

Z29 

.818 

.67 

71  OOO 

5550O 

.OOOO57 

.OO0092 

ii 

.828 

.68 

77400 

60500 

.O00052 

.000083 

" 

.611 

.76 

570OO 

44500 

.OO0058 

.000093 

Z37 

.619 

.76 

682OO 

5330O 

.OO0048 

.OOOO77 

.628 

.76 

7940O 

62OOO 

.OO0041 

.OOO065 

M 

.616 

.74 

81900 

64000 

.OOOO4O 

.000065 

Z41 

.623 

.76 

9200O 

7190O 

.000036 

.000057 

it 

.631 

.76 

102100 

798OO 

.OOO032 

.OOQ051 

M 

.619 

.73 

101000 

78900 

.OOO033 

.OOOO52 

Z45 

.626 

11020O 

86100 

.000030 

.OOOO48 

.633 

•76 

119500 

9330O 

.000027 

.000043 

M 

.519 

.83 

9000O 

70300 

.OOO031 

.000049 

Z53 

.526 

.83 

1O4900 

8190O 

.000026 

.OOOO42 

.632 

.84 

119700 

93500 

.O00023 

.OOOO36 

M 

.520 

.81 

123200 

96200 

.OOO022 

.OOOO36 

Z57 

.526 

.82 

136800 

106800 

.OOOO2O 

.000032 

.532 

.84 

15O400 

11750O 

.000018 

.O00029 

M 

.519 

.81 

149800 

117000 

.OOOO  18 

.00003O 

Z61 

.525 

.82 

16230O 

12680O 

.000017 

.000027 

.530 

.83 

17490O 

13670O 

.000015 

.OOOO25 

M 

.29 

.72 

10860O 

848OO 

.000020 

.000033 

Z67 

.26 

.71 

150800 

1178OO 

.OOOO  14 

.OOOO22 

Z73 

128  STRENGTH  OF  MATERIALS 

Average    Strength  in  Pounds  per  Square  Inch. 

TENSION -Ultimate 
Aluminum,  wire  45000 

Cement,  Portland  500 

Concrete,  28  days  250 

Copper,  drawn  *  35COO 

Fir  8000 

Iron,  cast  20000 

COMPRESSION-Ultimate 
Aluminum  12000 

Brick,  common  3000 

Brick,  pressed  8000 

Brickwork,  lime  mortar  1000 
Brickwork,  cement  mortar  2000 
Concrete,  28  days  2500 

Granite  14000 

SHEAR— Ultimate 

Britk  SOO 

Concrete,  28  days  250 

Iron,  cast  18000 

Iron,  wrought  45000 

Oak,  across  grain  4000 

COEFFICIENTS  OF  ELASTICITY 
Aluminum  11000000 

Concrete  2800000 

Copper  16000000 

Iron,  cast  18000000 

COLUMNS 


Iron,  wrought 
Leather,  belts 

50000 
3000 

Oak 

12000 

Pine 

7000 

Steel,  rivet 

55000 

Steel,  rolled 

65000 

Iron,  cast 

<)OCOO 

Iron,  wrought 

55000 

Marble 

10000 

Oak 

•  7000 

Pine,  fir 

5600 

Sandstone 

8000 

Steel,  rolled 

70000 

Oak,  with  grain 

800 

Pine  or  fir,  across  grain     2000 

Pine  or  fir,  with 

grain          400 

Steel,  rivet 

50000 

Steel,  rolled 

50000 

Iron,  wrought 
Steel 

25000000 
30000000 

Timber 

1500000 

Gordon's  Formula  :  P 


-r« 

2 

ar2 


where  P  is  the  ultimate  strength  in  pounds  per  square  inch 
and  L  is  the  length  in  feet;  r  is  the  least  radius  of  gyration  in 
inches  for  steel  columns  and  the  least  outside  diameter  or  side  in 
inches  for  round  and  rectangular  cast  iron  and  timber  columns. 
5  and  a  are  constants  given  as  follows  : 

Column 

Cast  iron,  square  end,  hollow  round 
Cast  iron,  square  end,  hollow  rectangular 
Medium  steel,  square  end 
Medium  steel,  pin  end 
Pine,  fir,  square  end 

For  structural  work,  the  safe  load  in  pounds  per  square  inch 
for  steel  columns  and  struts  may  be  taken  from  the  formula 

Ps  =  16000  —  840  — 

L  being  the  length  in  feet  and  r  the  least  radius  of  gyration  in 
inches. 


5 

80000 
80000 
50000 
50000 

5000 


a 
5 
7 

250 

125 

1  75 


BEAMS 


BEAMS 
Mechanics  of  Beams. 


For  a  simple  beam 
loaded  in  any  manner 
let  A  and  B  be  the 
reactions.  A  and  B 
are  easily  found  by 
the  principle  of  mo- 
ments; also  if  d  is 
the  distance  of  the 
center  of  mass  of  the 
load  P  from  A, 

A  = 


—  d)   andB  =         ' 


The  bending  moment  M  at  any  section  x—x  is  the  moment 
Ax  of  the  reaction  A  minus  the  integral  moment  j  z  dp  of  the 
load  between  A  and  % — x.  Hence, 


where  P*  is  the  load  between  A  and  x  —  x,  and  5  is  the  distance 
of  the  center  of  mass  of  Px  from  x  —  x. 

The  vertical  shear  Fat  any  section  x  —  x  is  the  reaction  A 
minus  the  sum  of  the  loads  between  A  and  x  —  x.  Also, 

y=    dJ*> 
dx 

and  hence,  when  F=  0,  M  is  a  maximum. 

For  a  moving  uniform  load  the  greatest  shear  at  any  section 
occurs  when  the  load  extends  from  the  end  of  the  span  to  tha* 
section;  the  maximum  bending  moment  at  any  section  occurs 
when  the  load  covers  the  entire  span. 


130  BEAMS 


Mechanics  of  Beams — Continued 

For  a  moving  system  of  concentrated  loads  the  center  of  the 
span  bisects  the  distance  between  the  point  of  maximum  bend- 
ing and  the  center  of  mass  of  the  loads. 

The  bending  moment  at  any  point  for  any  number  of  loads 
may  be  found  by  adding  algebraically  the  bending  moments  due 
to  each  of  the  loads  separately. 

If  p  is  the  allowable  unit  stress  for  the  material,  /the 
moment  of  inertia  of  the  beam  section  about  an  axis  through 
the  neutral  axis  of  the  beam  and  c  the  distance  of  the  most 
remote  fibre, 

M=  P-l  =  PS 

where  5  is  the  section  modulus. 

The  deflection  D  at  any  point  is  found  by  integrating 

»-«/-££• 

d  x2 

In  the  following  table  the  meaning  of  most  of  the  symbols 
is  made  clear  by  the  figures;  A  and  B  are  reactions,  m  =  bend- 
ing moment,  M  and  D  denote  maximum  bending  moments  and 
deflections  and  their  subscripts  denote  the  points  at  which  the 
maximum  occurs,  c  being  used  for  the  center  of  the  span,  /gives 
the  distance,  or  distances,  to  the  points  of  contraflecture  and 
both/ and  x  are  measured  from  the  left  reaction  A. 

Multiplying  the  load  for  any  given  case  by  X  gives  the  load 
which  will  produce  the  same  maximum  bending  moment  if  the 
beam  were  simply  supported  and  uniformly  loaded  over  the 
same  span  I. 

This  factor  will  be  found  convenient  in  connection  with 
tables  giving  the  strength  of  beams  uniformly  loaded  but,  ex- 
cepting in  the  case  of  beams  of  uniform  cross  section,  caution 
must  be  observed  since  the  maximum  bending  moments  for  the 
two  loadings  will  not  in  general  occur  at  the  same  point  of  the 
beam. 


BEAMS 


131 


Reactions,  Bending  Moment  and  Deflection  of  Beams 


jBetween  AandP'.m- 
Betwcen  PandB:m 


"*=™ 

Sab 

•*-    z« 


,  Mc=  PI 


Dc=  Pt+  48  ET  , 


Mc  -  Pa, 


£et ween  AanctP:    m=  Px 
two  loads :  m  =  Pc 

X=T 


pi    x       / 

Between  A  and  P :  m~  ~£~(T  ~~  ^) 

pi    X       j 
Between  P  and  B:  m  - —  (j  —  ;f) 


'/BZEI 


132 


BEAMS 


Reactions,  Bending  Moment  and  Deflection  of  Beams 


J3=P 


7n- 


A=  — 
*      or 


Mc 


77Z  = 

A- 


—(!-- 

2  u 


D 


P7  at    x= 


p 
O.OJ304-—  ,        z  =  a 


Pl 

T 


4 


BEAMS  133 

Reactions,  Bending  Moment  and  Deflection  of  Beams 


8 


m- 


f^B~  ~~Q~  -Absolute  maximum.       f~~2 
M=  —---Relative  maximum  at  *  =  ^ 


j)z= 


TV 


max.       A 
max. 


yl  = 


A 

n    PIJ           ,  . 

J84-EI                '     " 

Traveling 
Crane 


a  -max.  for-  X=  ~  and 


a_ 

/imax 

Mima* 

0 

Z.O      P 

O.JOOOOPl 

When  ^.686 

.7 

7.9      " 

.4-J7ZJ  " 

1 

.£ 

7.6       " 

.4-OSOO  " 

the  absolute 

.J 

77      » 

.36J2S  " 

value  of  Mjwa 

.4 
..J" 

7.<5      " 

/J"     " 

.3ZOOO  " 
.A67ZJ  " 

/j  always  PIA_, 
or  that  for  one 

.Jfi6 

7.W4" 

.ZSOOO  >' 

wheel  at  center. 

134  ATOMIC   WEIGHTS  AND   DENSITIES 

Valency  and  Atomic  Weight  of  Elements 


Element 


Symbol  Atomic          Atomic 

and  Weierht          Weight 

Valency  H=l  0=16 


Aluminum     Al  (3)  26.88 

Antimony    Sb  (3)  119.23 

Arsenic     As  (3)  74.4 

Barium     Ba  (2)  136.3 

Bismuth     Bi  (3)  206.83 

Boron    B  (2)  10.91 

Bromine     Br  (1)  79.32 

Cadmium     Cd  (2)  111.50 

Calcium    Ca  (2)  39.81 

Carbon    C  (4)  11.90 

Chlorine    Cl  (1)  35.17 

Chromium    Cr  (2)  (3)  51.68 

Cobalt     Co  (2)   (3)  58.53 

Copper     Cu  (1)  (2)  63.1 

Fluorine     F  (1)  18.85 

Gold    Au  (3)  195.62 

Hydrogen    H  (1)  1.00 

Iodine    1(1)  125.95 

Iridium     Ir  (2)  (4)  191.46 

Iron    Fe  (2)  (3)  55.44 

Lead     Pb  (2)  205.24 

Lithium     Li  (1)  6.97 

Magnesium     Mg  (2)  24.17 

Manganese      Mn  (2)  (3)  54.55 

Mercury     Hg  (1)  (2)  198.4 

Nickel    Ni  (2)  (3)  58.23 

Nitrogen    N  (3)  13.93 

Oxygen    O  (2)  15.87 

Palladium    Pd  (2)  105.15 

Phosphorus   P  (3)  30.75 

Platinum    Pt  (2)  (4)  193.24 

Potassium    K  (1)  38.84 

Selenium     Se  (2)  78.57 

Silicon     Si  (4)  28.17 

Silver     Ag  (1)  107.07 

Sodium    Nad)  22.87 

Strontium    Sr  (2)  86.9 

Sulphur     S  (2)  31.8 

Tellurium     Te  (2)  125.98 

Tin Sn  (2)  118.05 

Zinc    Zn  (2)  (4)  64.88 


120.2 

75.0 
137.4 
208.5 

11.0 

79.96 
112.4 

40.13 

12.0 

35.45 

52.1 

59.0 

63.6 

19.0 
197.2 

1.008 
126.97 
193.0 

55.9 
206.9 
7.03 

24.36 

55.0 
200.0 

58.7 

14.04 

16.0 
106.0 

31.0 
194. S 

39.15 

79.2 

28.4 
107.93 

23.05 

87.6 

32.06 
127.0 
119.0 

65.4 


Densities  of  Substances 


Wt.  of  1  cu.  ft.  of  Water  at 
4°C  or  39.1°Fz=62.4245  Ib. 


Air  at  0°C.  and 
Aluminum 
Antimony 
Asphaltum 

Brass,   cast 
Brass,  rolled 
Brick,   soft 
Brick,   common 

Brick,   hard 
Brick,   pressed 
Brick,   fire 


Grams 
per  cu.cm. 


Pounds 
per  cu.in 


.001293 
2.67 
6.76 
1.4 

8.1 
8.4 
1.6 

1.79 

2.0 

2.16 

2.3 


.0963 
.2439 


.2917 
.3031 


Pounds 
per  cu.ft. 


.0807 
166.5 
421.6 
87.3 

504 
524 
100 
112 

125 
135 
145 


DENSITIES   OF   SUBSTANCES 


135 


Wt.  of  1  cu.  ft.  of  Water  at 
4pCor39.1°F=  62.4245  Ib. 

Grams 
per  cu.  cm. 

Pounds 
per  cu-  in. 

Pounds 
per  cu.  ft. 

Cement,    Rosendale,   loose 
Cement,   Portland 
Coal,    anthracite 

.96 
1.25 
1.5 

60 
78 
93.5 

Coal,    anthracite,    loose 
Coal,    bituminous 
Coal,    bituminous,    loose 

1.3 

53 

84 
48 

Coke,   loose 
Concrete,    plain 
Concrete,   reinforced 

2.0 
2.4 

28 
125 
150 

Copper,    cast 
Earth,    dry,    loose 
Earth,   dry,  rammed 

8.85 

.3195 

552 

75 
95 

Glass 
Glass,    common   window 
Gold 

2.6 
2.52 
19.26 

.6949 

162 
157 
1200.9 

Granite 
Gravel 
Ice 

2.6 
1.75 
.92 

162 
109 
57.4 

Iridium 
Iron,   wrought 
Iron,    cast 

22.7 
7.7 
7.22 

.8076 
.2779 
.2604 

1396 
480 
450 

Lead 
Limestone,  marble 
Maple,    dry 

11.38 
2.6 

.4106 

709.7 
164 
49 

Masonry 
Mercury,    15  °C. 
Mica 

13.58 
2.93 

.490 

125-165 
846.8 
183 

Mud 
Nickel 
Oak,   dry 

8.8 

.3175 

115 
548.7 
60 

Petroleum 
Pine,    dry 
Fitch 

.878 
1.15 

54.8 
25-45 

72 

Platinum 
Salt 
Sand 

21.5 
1.1 

.7758 

1347 

68.6 
90  to  110 

Sandstone 
Silver 
Slate 

2.4 
10.51 
2.8 

.3791 

150 
655.1 
175 

Snow,  fresh  fallen 
Snow,  moistened 
Soapstone 

2.7 

5  to  12 
15  to  50 
170 

Steel 
Sulphur 
Tar 

7.85 
2.0 

1 

.2834 

489.6 
125 
62.4 

Tin* 
\vater,  4°C. 
Water,  100  °C. 

7.35 
1 
.958 

.2652 

458.3 
62.42 
59.83 

Water,   sea  water 
Zinc 

1.028 
7.0 

.2526 

64.08 
437 

136 


HEAT 


Comparison  of  Centigrade  and  Fahrenheit. 


c 

F 

C 

F 

C 

F 

C 

F 

C 

F 

—  40 

—40 

—17.8 

0 

4.4 

40 

27 

80.6 

49 

120.2 

—39.4 

—39 

—17.2 

1 

5 

41 

27.2 

81 

49.4 

121 

—  39 

—38.2 

—17 

1.4 

5.6 

42 

27.8 

82 

50 

122 

—38.9 

—38 

—16.7 

2 

6 

42.8 

28 

82.4  j 

50.6 

123 

—38.3 

—37 

—16.1 

3 

6.1 

43 

28.3 

83 

51 

123.8 

—38 

—36.4 

—16 

3.2 

6.7 

44 

28.9 

84 

51.1 

124 

—37.8 

—36 

—15.6 

4 

7 

44.6 

29 

84.2 

51.7 

125 

—37.2 

—35 

—15 

5 

7.2 

45 

30 

86 

52 

125.6 

—37 

—34.6 

—14.4 

6 

7.8 

46 

29.4 

85 

52.2 

126 

—36.7 

—34 

—14 

6.8 

8 

46.4 

30.6 

87 

52.8 

127 

—36.1 

—33 

—13.9 

7 

8.3 

47 

31 

87.8 

53 

127.4 

—36 

—32.8 

—13.3 

8 

8.9 

48 

31.1 

88 

53.3 

128 

—35.6 

—32 

—13 

8.6 

9 

48.2 

31.7 

89 

53.9 

129 

—  35 

—31 

—12.8 

9 

9.4 

49 

32 

89.6 

54 

129.2 

—34.4 

—30 

—12.2 

10 

10 

50 

32.2 

90   ' 

54.4 

130 

—34 

29.2 

—12 

10.4 

10.6 

.51 

32.8 

91 

55 

131 

—33.9 

—29 

—11.7 

11 

11 

51.8 

33 

91.4 

55.6 

132 

—33.3 

—28 

—11.1 

12 

11.1 

52 

33.3 

92 

56 

132.8 

—  33 

—27.4 

—11 

12  2 

11.7 

53 

33.9 

93 

56.1 

133 

—32.8 

—27 

—10.6 

13' 

12 

53.6 

34 

93.2 

56.7 

134 

39  2 

—26 

—10 

14 

12.2 

54 

34.4 

94 

57 

134.6 

—  32  " 

—25.6 

—  9.4 

15 

12.8 

55 

35 

95 

57.2 

135 

—31.7 

—25 

—  9 

15.8 

13 

55.4 

35.6 

96 

57.8 

136 

—31.1 

—24 

—  8.9 

16 

13.3 

56 

36 

96.8 

58 

136.4 

—  31 

—23.8 

-  8.3 

17 

13.9 

57 

36.1 

97 

58.3 

137 

—30.6 

—23 

—  8 

17.6 

14 

57.2 

36.7 

98 

58.9 

138 

—  30 

—22 

—  7.8 

18 

14.4 

58 

37 

98.6 

59 

138.2 

—29.4 

—21 

—  7.2 

19 

15 

59 

37.2 

99 

59.4 

139 

—29 

—  20.2 

—  7 

19.4 

15.6 

60 

37.8 

100 

60 

140 

—28.9 

—20 

—  6.7 

20 

16 

60.8 

38 

100.4 

60.6 

141 

—28.3 

—19 

—  6.1 

21 

16.1 

61 

38.3 

101 

61 

141.8 

—28 

—18.4 

—  6 

21  2 

16.7 

62 

38.9 

102 

61.1 

142 

—27.8 

—18 

—  5.6 

22 

17 

62.6 

39 

102.2 

61.7 

143 

—  97  2 

—17 

—  5 

23 

17.2 

63 

39.4 

103 

62 

143.6 

—27 

—16.6 

—  4.4 

24 

17.8 

64 

40 

104 

62.2 

144 

—  26.7 

—16 

—  4 

24.8 

18 

64.4 

40.6 

105 

62.8 

145 

—26.1 

—15 

—  3.9 

25 

18.3 

65 

41 

105.8 

63 

145.4 

—  26. 

—14.8 

—  3.3 

26. 

18.9 

66. 

41.1 

106. 

63.3 

146. 

—25.6 

—14. 

—  3. 

26.6 

19. 

66.2 

41.7 

107. 

63.9 

147. 

—25. 

—13. 

—  2.8 

27. 

19.4 

67. 

42. 

107.6 

64. 

147.2 

—24.4 

—12. 

—  2.2 

28. 

20. 

68. 

42.2 

108. 

64.4 

148. 

—24. 

—11.2 

—  2. 

28.4 

20.6 

69. 

42.8 

109. 

65. 

149. 

—23.9 

—11. 

-j  7 

29. 

21. 

69.8 

43. 

109.4 

65.6 

150. 

—  23.3 

—10. 

—  1.1 

30. 

21.1 

70. 

43.3 

110. 

66. 

150.8 

—  23. 

—  9.4 

—  1. 

30.2 

21.7 

71. 

43.9 

111. 

66.1 

151. 

—22.8 

—  9. 

—  0.6 

31. 

22.^ 

71.6 

44. 

111.2 

66.7 

152. 

22.2 

—  8. 

0 

32. 

72. 

44.4 

112. 

67. 

152.6 

—22." 

—  7.6 

0.6 

33. 

2  2".  8 

73. 

45. 

113. 

67.2 

153. 

21.7 

—  7. 

1. 

33.8 

23. 

73.4 

45.6 

114. 

67.8 

154. 

—  21.1 

—  6. 

1.1 

34. 

23.3 

74. 

46. 

114.8 

68. 

154.4 

—21. 

—  5.8 

1.7 

35. 

23.9 

75. 

46.1 

115. 

68.3 

155. 

—20.6 

—  5. 

2. 

35.6 

24. 

75.2 

46.7 

116. 

68.9 

156. 

—20. 

—  4. 

2.2 

36. 

24.4 

76. 

47. 

116.6 

69. 

156.2 

—19.4 

—  3. 

2.8 

37. 

25. 

77. 

47.2 

117. 

69.4 

157. 

—19. 

—  2.2 

3. 

37.4 

25.6 

78. 

47.8 

118. 

70. 

158. 

18.9 

—  2. 

3.3 

38. 

26. 

78.8 

48. 

118.4 

70.6* 

159. 

—  18  3 

—  1. 

3.9 

39. 

26.1 

79. 

48.3 

119. 

71. 

159.S 

—18. 

—  0.4 

4. 

39.2 

26.7 

80. 

48.9 

120. 

71.1 

160. 

HEAT 


137 


Comparison  of  Centigrade  and  Fahrenheit. 


c 

F            C 

F              C 

F 

C 

F 

C 

F 

71.7 

161.      !     93.9 

201.           116.1 

241. 

138.3 

281. 

161. 

321.  S 

72. 

161.6         94. 

201.2         116.7 

242. 

138.9 

282. 

161.1 

322. 

72.2 

162.          94.4 

202.           117. 

242.6 

139. 

282.2 

161.7 

323. 

72.8 

163. 

95. 

203.           117.2 

243. 

139.4 

283. 

162. 

323.6 

/3. 

163.4 

95.6 

204.           117.8 

244. 

140. 

284. 

162.2 

324. 

73.3 

164. 

96. 

204.8         118. 

244.4 

140.6 

285. 

162.8 

325. 

73.9 

165. 

96.1 

205.           118.3 

245. 

141., 

285.8         163. 

325.4 

74. 

165.2         96.7 

206.           118.9 

246. 

141.1 

286.     !       163.3 

326. 

74.4 

166.           97. 

206.6         119. 

246.2 

141.7 

287.           163.9 

327. 

75. 

167. 

97.2 

207.           119.4 

247. 

142. 

287.6         164. 

327.2 

75.6 

168. 

97.8 

208.           120. 

248. 

142.2 

288.           164.4 

328. 

76. 

168.8 

98. 

208.4  ;      120.6 

249. 

142.8 

289.           165. 

329. 

76.1 

169. 

98.3 

209.     ;      121. 

249.8 

143. 

289.4         165.6 

330. 

76.7 

170. 

98.9 

210.           121.1 

250. 

143.3 

290.           166. 

330.8 

77. 

170.6 

99. 

210.2         121.7 

251. 

143.9 

291.           166.1 

331. 

77.2 

171. 

99.4 

211.           122. 

251.6 

144. 

29121       166.7 

332. 

77.8 

172. 

100. 

212.           122.2 

252. 

144.4 

292!           167. 

332.6 

78. 

172.4       100.6 

213.      '      122.8 

253. 

145. 

293.           167.2 

333. 

78.3 

173. 

101. 

213.8         123. 

253.4 

145.6 

294.           167.8 

334. 

78.9 

174. 

101.1 

214.           123.3 

254. 

146. 

294.8         168. 

334.4 

79. 

174.2 

101.7 

215.     !      123.9 

255. 

146.1 

295.           168.3 

335. 

79.4 

175. 

102. 

215.6  .      124. 

•  255.2 

146.7 

296.           168.9 

336. 

80. 

176. 

102.2 

216.           124.4 

256. 

147. 

296.  6  *       169. 

336.2 

80.6 

177. 

102.8 

217.           125. 

257. 

147.2 

297.           169.4 

337. 

81. 

177.8 

103. 

217.4  !      125.6 

258. 

147.8 

298.           170. 

338. 

81.1 

178. 

103.3 

218.           126. 

258.8 

148. 

298  4         170.6 

339. 

81.7 

179. 

103.9 

219. 

126.1 

259. 

148.3 

299^           171. 

339.8 

82. 

179.6 

104. 

219.2 

126.7 

260. 

148.9 

300. 

171.1 

340. 

82.2 

180.      !   104.4 

220.           127. 

260.6 

149. 

300.2 

171.7 

341. 

82.8 

181.         105. 

221. 

127.2 

261. 

149.4 

301. 

-  172. 

341.6 

83. 

181.4       105.6 

222. 

127.8 

262. 

150. 

302. 

172.2 

342. 

83.3 

182.      i   106. 

222.8 

128. 

262.4 

150.6 

303. 

172.8 

343. 

83.9 

183.      '  106.1 

223. 

128.3 

263. 

151. 

303.8 

173. 

343.4 

84. 

183.2       106.7 

224. 

128.9 

264. 

151.1 

304. 

173.3 

344. 

84.4 

184.         107. 

224.6 

129. 

264.2 

151.7 

305. 

173.9 

345. 

85. 

185.         107.2 

225. 

129.4 

265. 

152. 

305.6 

174. 

345.2 

85.6 

186.         W8 

226. 

130. 

266. 

152.2 

306. 

174.4 

346. 

£6. 

186.8       108. 

226.4 

130.6 

267. 

152.8 

307. 

175. 

347. 

86.1 

187.      |   108.3 

227. 

131. 

267.8 

153. 

307.4         175.6 

348. 

86.7 

188.      !  108.9 

228. 

131.1 

268. 

153.3 

308.           176. 

348.8 

87. 

188.6 

109. 

228.2 

131.7 

269. 

153.9 

309. 

176.1 

349. 

87.2 

189. 

109.4 

229. 

132. 

269.6 

154. 

309.2 

176.7 

350. 

87.8 

190.         110. 

230. 

132.2 

270. 

154.4 

310. 

177. 

350.6 

88. 

190.4 

110.6 

231. 

132.8 

271. 

155. 

311. 

177.2 

351. 

88.3 

191. 

111. 

231.8 

133. 

271.4 

155.6 

312. 

177.8 

352. 

88.9 

192.         Hl.l 

232. 

133.3 

272. 

156. 

312.8 

178. 

352.4 

89. 

19->  9.      111.7 

233. 

133.9 

273. 

156.1 

313. 

178.3 

353. 

89.4 

193.         112. 

233.6 

134. 

273.2 

156.7 

314. 

178.9 

354. 

90. 

194.         H2.2 

234. 

134.4 

274. 

157. 

314.6 

179. 

354.2 

90.6 

195.      i   112.8 

235. 

135. 

275. 

157.2 

315. 

179.4 

355. 

91. 

195.8    !   113. 

235.4 

135.6 

276. 

157.8 

316. 

180. 

356. 

91.1 

196.      i   H3.3 

236. 

136. 

276.8 

158. 

316.4 

180.6 

357. 

91.7 

197.         113.9 

237. 

136.1 

277. 

158.3 

317. 

181. 

357.8 

92. 

197.6       114. 

237.2 

136.7 

278. 

158.9 

318. 

181.1 

358. 

92.2 

198.         H4.4 

238. 

137. 

278.6 

159. 

318.2 

181.7 

359. 

92.8 

199.         115. 

239. 

137.2 

279. 

159.4 

319. 

182. 

359.6 

93. 

199.4    i  115.6 

240. 

137.8 

280. 

160. 

320. 

182.2 

360. 

200.         116. 

240.8 

138. 

280.4 

160.6 

321. 

182.8 

361. 

138 


HEAT 


Coefficients  of  Expansion  and  Specific  Heat. 


Solids. 
Coefficient  x  107 


1°  F 

1°  C 

Aluminum 

123 

222 

Antimony 

63 

113 

Brass,  cast 

96 

172 

Brass,  plate 

105 

189 

Bronze 

99 

177 

Bismuth 

97 

176 

Cement,  Portland 

59 

107 

Concrete 

80 

143 

Copper 

89 

160 

Glass,  flint 

45 

81 

Glass,  thermometer 

50 

90 

Glass,  hard 

40 

71 

Gold 

79 

142 

Ice 

Iridium 

36 

64 

Iron,  cast 

56 

100 

Iron,  wrought 

65 

117 

Lead 

157 

283 

Nickel 

70 

125 

Plaster 

92 

166 

Platinum 

48 

86 

Platinum  85%  } 
Iridium     15%  f 

45 

82 

Porcelain 

20 

36 

Silver 

108 

194 

Steel,  cast 

64 

114 

Steel,  tempered 

69 

124 

Sulphur 

Tin 

116 

209 

Wood,  pine 

28 

50 

Zinc 

141 

253 

Liquids 

(Cu. 

Expansion.) 

Alcohol 

7780 

14000 

Glycerine 
Olive  Oil 

2780 
4110 

5000 
7400 

Petroleum 

5530 

10000 

Mercury 

1011 

1820 

Turpentine 

5530 

10000 

Specific 

Heat 

.21 

.05 

\  .094 


.093 
.20 

.032 
.502 

.130 

.031 
.11 

.033 


.055 
.117 

.18 
.056 

.094 

.58 
.58 
.40 
.50 
.033 
.42 


The  specific  heat  of  water  for  t  in  degrees  cent,  is 
c  =    1  +  .00004  t  +  .0000009  t\ 


HEAT 

Density  and  Volume  of  Water. 


139 


Temp. 
C      F 

Density 

Volume 

Temp. 
C     F 

Density 

Volume 

0 

32 

.99987 

1.00013 

60 

140 

.9833 

1.0170 

2 

35.6 

.99997 

1.00003 

80 

176 

.9719 

1.0289 

4 

39.2 

1.00000 

1.00000 

100 

212 

.9586 

1.043-4 

6 

42.8 

.99997 

1.00003 

120 

248 

.9435 

1.0599 

10 

50. 

.99973 

1.00027 

140 

284 

.9263 

1.0795 

15 

59. 

.99913 

1.00087 

160 

320 

.9076 

1.1018 

24 

75.2 

.99733 

1.00268 

180 

356 

.8875 

1.1268 

40 

104 

.99233 

1.00773 

200 

392 

.8662 

1.1544 

Melting  and  Boiling  Points. 

At  760  mm.  Mercury. 


Substance 

Melting 
Cent. 

or  Freezing          Boiling 
Fahr.       Cent.  Fahr. 

Alcohol,  absolute 
Aluminum 
Ammonia  gas 
Antimony 

—  100 
625 

—     78 
430 

—  148         78.5 
1157 
—  108        —33 
806 

173 

27 

Brass 
Bronze 
Copper 
Glass 

900 

900 
1084 
800—1400 

1652 
1652 
1983 
1470—2500 

Gold 
Iridium 
Iron,    cast 
Iron,    wrought 

1064 
1950 
1150 
1550 

1945 
3540 
2100 

2820 

Lead 

Mercury 
Nickel 
Paraffine 

327 
—39 
1470 
54 

620 
—38.2          357 
2680 
129          300 

675 

570 

Platinum            « 
Silver 
Steel 
Sulphur 

1800 
962 
1350 
115 

3270 
1760 
2460 
239           445 

833 

Sulphur    Dioxide 
Tin 
Zinc 

—76 
232 
420 

—105        —10 
450 
790          915 

14 
1680 

Boiling  Point  of  Water 
for  various  pressures  in  mm.  of  Mercury. 

Pressure.       Cent 

Pressure.        Temp. 
Cent. 

Pressure. 

Temp. 
Cent. 

680                      96.92 
690                      97.32 
700                       97.71 
710                       98.11 
720                       98.49 

730                   98.88 
740                   99.26 
750                    99.63 
755,                   99.82 
760                  100.00 

765 
770 
780 
790 
800 

100.18 
100.37 
100.73 
101.09 
101.44 

140 


HEAT 


Latent  Heat. 


Latent  Heat 
760  mm.  Mercury 

Substance 

Fusion 
Calories              B.  T.  U. 

Vaporization 
Calories       B  .  T.  U 

Alcohol 
Lead 
Mercury 

6 

2.8 

10.8 
5. 

210                      378 
62                  111.6 

Platinum 
Silver 
Sulphur 

27 
21 
9 

48.6 
37.8 
16.2 

362                      652 

Tin 
Water   (Ice)    (Steam) 
Zinc 

13 
80.1 
28 

23.4 

144.2 
50.4 

537                 965.  S 

Gases. 

Boyle's  Law:      Pv  = 

constant. 

Charles'  Law  :      P  = 

PO    (1    +         '-    ' 

\-  r. 

for  cent  temp.; 

273 


P'   (1 


32\  =  P/   T' 
A  )      491.4 


forFahr.temp. 


273 

t'  —  32 

491 

t  =  temp,  in  deg.  cent.,     t'  —  temp,  in  deg.  Fahr. 
T  =  273  +  t  and  T'  =  459.4  -r  t' 

P  =  pressure  at  t  or  t',     P0  =  pressure  at  0°  C, 
P'  =  pressure  at  32°  F. 

Combining  these  two  laws  gives  the  equation, 
P  v  =  R  T  for  metric  units  or  Pv  =  E'T'  for  English; 

in  which  the  gas  constant  R  or  R]  is  inversely  proportional  to 
the  density,  or  to  the  molecular  weight  m  of  the  gas;  if  the 
molecular  weight  of  oxygen  is  taken  as  32,  and  r0  =  volum  e 
at  0°C  and  v1  =  volume  at  32°  F,  then 


R  = 


273 


847 
m 


and 


/»  t;1    =  1541.5 
491.4  m 


HEAT  141 

Density  and  Constant  of  Gases. 


Gas 

Density 
Air  =  l 

*  Weight 
of  1                of  1 
cu.  m.           cu.  ft. 

Constant 
Metric        English 
units            units 
It                W 

Air 
Alcohol 
Ammonia 
Carbon  Dioxide 
Hydrogen 
Oxygen 
Steam 
Sulphur  Dioxide 

1.000 
1.590 
.590 
1.520 
.0696 
1.105 
.623 
2.213 

1.295 
2.059 
.763 
1.966 
.090 
1.430 
.806 
2.863 

.8073 
1.284 
.476 
1.227 
.0562 
.892 
.503 
1.787 

29.26 
18.4 
49.6 
19.25 
420 
26.5 
47.0 
13.2 

53.27 
33.5 
90.27 
35.03 
764 
48.2 
85.54 
24.  0.5 

*    At  0°  C  =  32°  F  and  760  mm.  mercury,  or  14.697  Ib.  per  sq.  in. 

Specific  Heat  of  Gases. 


Gas 

Specific 

Constant 
Pressure 
Cp 

Heat 

Constant                  -vP- 
Volume                     Cv 
Cv 

Air 
Alcohol 
Ammonia 
Carbon    Dioxide 
Hydrogen 
Oxygen 
Steam 
Sulphur    Dioxide 

.238 
.45 
.53 
.21 
3.41 
'    .217 
.50 
.15 

.170                         1.40 
.40                             .14 
.41                              .28 
.16                              .28 
2.42                              .40 
.155                            .40 
.39                              .28 
.12                              .25 

Factor  of  Evaporation. 

The  Factor  of  Evaporation     F  ==    -    ~  ~  h 

965.8 

where  H  is  the  total  heat  of  the  steam  at  the  given 
pressure  and  h  is  the  heat  of  the  liquid  at  the  given  temper- 
ature. 

In  the  following  table,  three  decimals  are  given  which  is 
usually  sufficient.  If  very  great  accuracy  is  required  F  must 
be  found  by  the  above  formula,  taking  the  values  of  H  and  h 
from  the  table  for  saturated  steam.  F  may  be  found  very 
accurately  to  four  decimals  by  the  following  formula. 

Ft  =  F32  —  0.001035  (t  —  32); 

in  which  Ft  is  the  factor  for  any  given  pressure  of  steam  with 
feed  water  at  t°  Fahr.  temperature  and  jP32  is  the  factor  for 
the  same  steam  pressure  with  feed  water  at  32°  Fahr. 


142 


HEAT 
Factors  of  Evaporation. 


Gauge 
Pressure 
Lbs.          32 

Temperature   of   Feed   Water.     Deg-.    Fahr. 
40            50             60             70             80             90           100 

110 

120 

0 

1.1873 

1.179 

1.169 

1.158 

1.148 

1.138 

1.127 

1.117 

1.107 

1.096 

10 

1.1958 

1.188 

1.177 

1.167 

1.157 

1.146 

1.136 

1.125 

1.115 

1.105 

20 

1.2019 

1.194 

1.183 

1.173 

1.163 

1.152 

1.142 

1.132 

1.121 

1.111 

30 

1.2068 

1.199 

1.188 

1.178 

1.168 

1.157 

1.147 

1.136 

1.126 

1.116 

40 

1.2107 

1.203 

1.192 

1.182 

1.171 

1.161 

1.151 

1.140 

1.130 

1.120 

50 

1.2141 

1.206 

1.196 

1.185 

1.175 

1.164 

1.154 

1.144 

1.133 

1.123 

60 

1.2172 

1.209 

1.199 

1.188 

1.178 

1.167 

1.157 

1.147 

1.136 

1.126 

70 

1.2199 

1.212 

1.201 

1.191 

1.181 

1.170 

1.160 

1.150 

1.139 

1.129 

80 

1.2224 

1.214 

1.204 

1.193 

1.183 

1.173 

1.162 

1.152 

1.142 

1.131 

90 

1.2247 

1.216 

1.206 

1.196 

1.185 

1.175 

1.165 

1.154 

1.144 

1.133 

100 

1.2269 

1.219 

1.208 

1.198 

1.188 

1.177 

1.167 

1.156 

1.146 

1.136 

110 

1.2288 

1.221 

1.210 

1.200 

1.189 

1.179 

1.169 

1.158 

1.148 

1.138 

120 

1.2307 

1.223 

1.212 

1.202 

1.191 

1.181 

1.171 

1.160 

1.150 

1.140 

130 

1.2322 

1.224 

1.214 

1.203 

1.193 

1.183 

1.172 

3.162 

1.152 

1.141 

140 

1.2341 

1.226 

1.215 

1.205 

1.194 

1.184 

1.174 

1.164 

1.153 

1.143 

150 

1.2358 

1.227' 

1.217 

1.207 

1.196 

1.186 

1.176 

1.165 

1.155 

1.145 

160 

1.2372 

1.229 

1.218 

1.208 

1.197 

1.187 

1.177 

1.167 

1.156 

1.146 

170 

1.2386 

1.230 

1.220 

1.210 

1.199 

1.189 

1.179 

1.168 

1.158 

1.147 

180 

1.2401 

1.232 

1.221 

1.211 

1.200 

1.190 

1.180 

1.170 

1.159 

1.149 

190 

1.2415 

1.233 

1.223 

1.212 

1.202 

1.192 

1.181 

1.171 

1.160 

1.150 

200 

1.2427 

1.234 

1.224 

1.214 

1.203 

1.193 

1.183 

1.172 

1.162 

1.151 

250 

1.2485 

1.240 

1.230 

1.219 

1.209 

1.199 

1.188 

1.178 

1.168 

1.157 

300 

1.2535 

1.245 

1.235 

1.224 

1.214 

1.204 

1.193 

1.183 

1.173 

1.162 

Gauge 
Pressure 
Lbs.         130 

Temperature   of   Feed   Water. 
140           150           160           170          180 

Deg.    Fahr. 
190         200 

210 

212 

0 

1.086 

1.075 

1.065 

L.055 

1.045 

1.034 

1.024 

1.013 

1.003 

1.000 

10 

1.095 

1.084 

1 

.074       1.063 

1.053 

1.043 

1.032 

1.022 

1.012 

1.009 

20 

1.101 

1.090 

1 

.080       1.069 

1.059 

1.049 

1.038 

1.028 

1.018 

1.015 

30 

1.106 

1.095 

1 

.085       1.074 

1.064 

1.054 

1.043 

1.033 

1.023 

1.020 

40 

1.110 

1.099 

1 

.089       1.078 

1.068 

1.058 

1.047 

1.037 

1.027 

1.024 

50 

1.113 

1.102 

1 

.091       1.081 

1.070 

1.060 

1.050 

1.039 

1.029 

1.027 

60 

1.116 

1.105 

1 

.095       1.084 

1.074 

1.063 

1.053 

1.043 

1.032 

1.030 

70 

1.118 

1.108 

1 

.098       1.087 

1.077 

1.066 

1.056 

1.045 

1.035 

1.033 

80 

1.121 

1.110 

1 

.100       1.090 

1.079 

1.069 

1.058 

1.048 

1.037 

1.035 

90 

1.123 

1.113 

1 

102 

.092 

1.081 

1.071 

1.060 

1.050 

1.040 

1.038 

100 

1.125 

1.115 

1 

.104 

.094 

1.083 

1.073 

1.063 

1.052 

1.042 

1.040 

110 

1.127 

1.117 

1 

.106 

.096 

1.085 

1.075 

1.065 

1.054 

1.044 

1.042 

120 

1.129 

1.119 

1 

108 

.098 

1.087 

1.077 

1.066 

1.056 

1.046 

1.044 

130 

1.130 

1  120 

1 

110 

.100 

1.089 

1.079 

1.068 

1.058 

1.047 

1.045 

140 

1.132 

1.122 

1 

111 

.101 

1.091 

1.080 

1.070 

1.059 

1.049 

1.047 

150 

1.134 

1.124 

1 

113 

.103 

1.092 

1.082 

1.071 

1.061 

1.051 

1.049 

160 

1.136 

1.125 

1 

115 

.104 

1.094 

1.083 

1.073 

1.063 

1.052 

1.050 

170 

1.137 

1.127 

1 

116 

.106 

1.095 

1.085 

1.074 

1.064 

1.053 

1  051 

180 

1.138 

1.128 

1 

118       ] 

.107 

1.097 

1.086 

1.076 

1.065 

1.055 

1.053 

190 

1.140 

1.129 

1 

119       1.108 

1.098 

1.088 

1.077 

1.067 

1.056 

1.054 

200 

1.141 

1.131 

1 

120       1.110 

1.099 

1.089 

1.078 

1.068 

1.057 

1.055 

250 

1.147 

1.136 

1 

126       1.116 

1.105 

1.095 

1.084 

1.074 

1.063 

1.061 

300 

1.152 

1.141 

1 

131       1.121 

1.110 

1.100 

1.089 

1.079 

1.068 

1.066 

SATURATED   STEAM 


143 


Properties  of  Saturated  Steam 


Abso- 
lute    Tcm- 
Press  perature 
Lbs.        Des. 
per       Fah. 
Sq.   in. 

Heat 

Volume 

Relative,     Specific 
Water         Cu.  ft. 
at             per  Ib. 
?9°F. 

Density 
Wt.   of 

1  CU.  ft. 

Lbs. 

Total 

of 

Liquid 

of 

Vapor- 
ization 

of 

Internal 
Work 

of 
External 
Work 

Entropy 
of 
Liquid 

0.10 

35.0 

1092.6 

3.0 

1089.6 

1033.6 

56.0 

.0061 

188724 

3022 

.00033 

0.15 

45.5 

1095.8 

13.6 

1082.3 

1025.4 

56.9 

.0272 

127023 

2050 

.00051 

0.20 

53.3 

1098.2 

21.4 

1076.8 

1019.3 

57.5 

.0426 

97110 

1555 

.00064 

0.30 

64.7 

1101.7 

32.8 

1068.9 

1010.3 

58.6 

.0645 

65760 

1053 

.00095 

0.40 

73.1 

1104.2 

41.2 

1063.0 

1003.7 

59.3 

.0804 

50397 

800.7 

.00125 

0.50 

79.8 

1106.2 

47.9 

1058.3 

998.4 

59.9 

.0929 

40411 

647.1 

.00155 

0.60 

85.5 

1108.0 

53.5 

1054.5 

994.1 

60.4 

.1031 

33973 

544.0 

.00184 

0.80 

94.6 

1110.8 

62.6 

1048.2 

986.9 

61.2 

.1197 

25842 

413.8 

.00242 

1.00 

102.0 

1113.1 

70.0 

1043.0 

981.1 

61.9 

.1329 

20896 

334.6 

.00299 

1.25 

109.6 

1115.4 

77.6 

1037.8 

975.2 

62.7 

.1462 

16549 

265.0 

.00369 

150 

115.9 

1117.3 

83.9 

1033.4 

970.1 

63.3 

.1573 

14226 

227.8 

.00438 

1.75 

121.4 

1119.0 

89.5 

1029.4 

965.7 

63.7 

.1669 

12303 

197.0 

.00508 

2.0 

126.3 

1120.5 

94.4 

1026.1 

961.9 

64.2 

.1754 

10841 

173.6 

.00576 

2.5 

134.6 

1123.0 

102.8 

1020.4 

955.1 

65.1 

.1895 

8780 

140.6 

.00711 

3.0 

141.6 

1125.1 

109.8 

1015.3 

949.5 

65.8 

.2013 

7394 

118.4 

.00844 

3.5 

147.7 

1127.0 

116.0 

1010.0 

944.6 

66.4 

.2114 

6395 

102.4 

.00976 

4.0 

153.1 

1128.6 

121.4 

1007.2 

940.4 

66.8 

.2203 

5640 

90.31 

.01107 

4.5 

158.0 

1130.1 

126.4 

1003.7 

936.4 

67.3 

.2284 

5039 

80.70 

.01239 

5.0 

162.3 

1131.5 

130.7 

1000.8 

933.1 

67.7 

.2353 

4573 

73.22 

.01366 

•5.5 

166.4 

1132.7 

134.8 

997.9 

929.8 

68.1 

.2419 

4180 

66.93 

.01494 

6.0 

170.1 

1133.8 

138.6 

995.2 

926.7 

68.5 

.2480 

3851 

61.67 

.01622 

6.5 

173.6 

1134.9 

142.1 

992.8 

924.0 

68.8 

.2535 

3573 

57.21 

.01748 

7.0 

176.9 

1135.9 

145.4 

990.5 

921.4 

69.1 

.2587 

3333 

53.37 

.01874 

7.5 

180.0 

1136.8 

148.5 

988.3 

918.9 

69.4 

.2636 

3123 

50.01 

.02000 

8.0 

182.9 

1137.7 

151.5 

986.2 

916.5 

69.7 

.2682 

2939 

47.07 

.02125 

8.5 

185.7 

1138.6 

154.3 

984.3 

914.4 

70.0 

.2725 

2773 

44.41 

.02250 

9.0 

188.3 

1139.4 

156.9 

982.5 

912.4 

70.1 

.2766 

2631 

42.13 

.02374 

9.5 

190.8 

1140.1 

159.4 

980.7 

910.3 

70.4 

.2805 

2501 

40.05 

.02497 

10.0 

193.2 

1140.9 

161.9 

979.0 

908.4 

70.6 

.2842 

2383 

38.16 

.02621 

10.5 

195.6 

1141.6 

164.3 

977.3 

906.5 

70.8 

.2878 

2276 

36.45 

.02743 

11.0 

197.8 

1142.3 

166.5 

975.8 

904.8 

71.0 

.2912 

2178 

34.88 

.02866 

11.5 

200.0 

1142.9 

168.7 

974.2 

903.0 

71.2 

.2946 

2086 

33.40 

.02990 

12.0 

202.0 

1143.6 

170.7 

972.9 

901.5 

71.4 

.2976 

2007 

32.14 

.03111 

12.5 

204.0 

1144.2 

172.7 

971.5 

900.0 

71.5 

.3007 

1931 

30.92 

.03234 

13.0 

205.9 

1144.7 

174.6 

970.1 

898.4 

71.7 

.3035 

1862 

29.82 

.03355 

13.5 

207.8 

1145.4 

176.7 

968.6 

896.8 

71.9 

.3066 

1796 

28.76 

.03478 

14.0 

209.6 

1145.8 

178.3 

967.5 

895.5 

72.0 

.3091 

1735 

27.79 

.03600 

14.5 

211.3 

1146.1 

180.1 

966.3 

894.2 

72.2 

.3117 

1680 

26.91 

.03677 

15 

213.0 

1146.9 

181.8 

965.1 

892.6 

72.5 

.3143 

1633 

26.15 

.03826 

16 

216.3 

1147.9 

185.1 

962.8 

890.0 

72.8    „ 

.3192 

1536 

24.59 

.04067 

144 


SATURATED  STEAM 


Properties  of  Saturated  Steam 


Abso- 
lute    xTem- 
Press.  perature 
Lbs.      Deg.         Total 
per       Fah. 
Sq.  in. 

Heat 

of                 of 
Liquid      Vapor- 
ization 

of 
Internal 
Work 

of 
External 
Work 

Entropy 
of 
Liquid 

Volume 
Relative.  Specific. 
Water     Cu.  ft. 
at          per  Ib. 
39°F 

Density 
Wt.  of 
1  cu.  ft. 

Lbs. 

17 

219.4 

1148.9 

188.3 

960.6 

887.6 

73.0 

.3238 

1451 

23  22 

.04307 

18 

222.4 

1149.8 

191.3 

958.5 

885.3 

73.2 

.3282 

1375 

22.00 

.04547 

19 

225.2 

1150.7 

194.1 

956.6 

883.2 

73.4 

.3324 

1305 

20.90 

.04786 

20 

227.9 

1151.5 

196.9 

954.6 

881.0 

73.6 

.3363 

1243 

19.91 

.05023 

21 

230.5 

1152.3 

199.5 

952.8 

879.0 

73.8 

.3401 

1187- 

19.01 

.05259 

22 

233.1 

1153.0 

202.0 

951.0 

877.0 

74.0 

.3438 

1137 

18.20 

.05495 

23 

235.5 

1153.7 

204.5 

949.2 

875.0 

74.2 

.3473 

1090 

17.45 

.05731 

24 

237.8 

1154.4 

206.8 

947.6 

873.2 

74.4 

.3506 

1047 

16.76 

.05966 

25 

240.0 

1155.1 

209.1 

946.0 

871.5 

74.5 

.3539 

1007 

16.13 

.06199 

26 

242  2 

1155.8 

211.2 

944.6 

869.9 

74.7 

.3570 

971.1 

15.55 

.06432 

27 

244.3 

1156.5 

213.4 

943.1 

868.2 

74.9 

.3600 

936.7 

15.00 

.06666 

28 

246.3 

1157.1 

215.4 

941.7 

866.7 

75.0 

.3629 

904.9 

14.49 

.06899 

29 

248.3 

1157.7 

217.4 

940.3 

865.1 

75.2 

.3657 

876.2 

14.03 

.07130 

30 

250.3 

1158.3 

219.4 

938.9 

863.6 

75.3 

.3685 

848.7 

13.59 

.07360 

31 

252.2 

1158.8 

221.3 

937.5 

862.0 

75.5 

.3712 

823.1 

13.18 

.07590 

32 

254.0 

1159.4 

223.1 

936.3 

860.7 

75.6 

.3737 

798.1 

12.78 

.07821 

33 

255.8 

1159.9 

224.9 

935.0 

859.2 

75.8 

.3762 

775.0 

12.41 

.08051 

34 

257.5 

1160.4 

226.7 

933.7 

857.8 

75.9 

.3787 

753.8 

12.07 

.08280 

35 

259.2 

1161.0 

228.4 

932.6 

856.6 

76.0 

.3811 

733.8 

11.75 

.08508 

36 

260.9 

1161.5 

230.0 

931.5 

855.3 

76.2 

.3834 

715.0 

11.45 

.08736 

37 

262.5 

1162.0 

231.7 

930.3 

854.0 

76.3 

.3856 

697.0 

11.16 

.08964 

38 

264.1 

1162.5 

233.3 

929.2 

852.8 

76.4 

.3878 

679.4 

10.88 

.09191 

39 

265.6 

1163.0 

234.8 

928.2 

851.7 

76.5 

.3900 

663.2 

10.62 

.09417 

40 

267.1 

1163.4 

236.4 

927.0 

850.3 

76.7 

.3921 

647.6 

10.37 

.09644 

41 

268.6 

1163.9 

237.9 

926.0 

849.2 

76.8 

.3942 

632.6 

10.13 

.09869 

42 

270.1 

1164.3 

239.3 

925.0 

848.1 

76.9 

.3962 

618.9 

9.91 

.1009 

43 

271.5 

1164.8 

240.8 

924.0 

847.0 

77.0 

.3982 

605.1 

9.69 

.1032 

44 

272.9 

1165.2 

242.2 

923.0 

845.9 

77.1 

.4001 

592.0 

9.48 

.1054 

45 

274.3 

1165.6 

243.6 

922.0 

844.8 

77.2 

.4020 

580.2 

9.29 

.1077 

46 

275.7 

1166.0 

245.0 

921.0 

843.7 

77.3 

.4038 

568.3 

9.10 

.1099 

47 

277.0 

1166.4 

246.3 

920.1 

842.7 

77.4 

.4056 

556.4 

8.91 

.1122 

48 

278.3 

1166.8 

247.6 

919.2 

841.7 

77.5 

.4074 

545.8 

8.74 

.1144 

49 

279.6 

1167.2 

248.9 

918.3 

840.7 

77.6 

.4092 

535.2 

8.57 

.1166 

50 

280.9 

1167.6 

250.2 

917.4 

839.7 

77.7 

.4109 

525.2 

8.41 

.1188 

51 

282.1 

1168.0 

251.5 

916.5 

838.7 

77.8 

.4126 

515.8 

8.26 

.1211 

52 

283.3 

1168.4 

252.7 

915.7 

837.8 

77.9 

.4143 

506.5 

8.11 

.1233 

53 

284.5 

1168.7 

253.9 

914.8 

836.8 

78.0 

.4160 

497.7 

7.97 

.1255 

54 

285.7 

1169.1 

255.1 

914.0 

835.9 

78.1 

.4175 

489.0 

7.83 

.1277 

55 

286.9 

1169.4 

256.3 

913.1 

834.9 

^  78  2 

.4191 

480.9 

7.70 

.1299 

56. 

288.0 

1169.8 

257.5 

912.3 

834.0 

78.3 

.4207 

472.7 

7.57 

.1321 

SATURATED   STEAM 


145 


Properties  of  Saturated  Steam 


Abso- 
lute   Tem- 
3ress.  perature 
Lbs.    Deg. 
per     Fah. 
3q.  in 

Heat 

Entropy 
of 
Liquid 

Volume 
Relative.     Specific.  Density. 
Water        Cu.  ft.      Wt.  of 
at             per  Ib.     1  cu.  ft. 
39°F.                       Lbs. 

Total 

of 
Liquid 

of 

Vapor- 
ization 

Internal  External 
Work      Work 

57 

289.2 

1170.1 

258.6 

911.5 

833.1 

78.4 

4222 

464.6 

7.44 

.1344 

58 

290.3 

1170.5 

259.7 

910.8 

832.4 

78.4 

!4237 

457.1 

7.32 

.1366 

59 

291.4 

1170.8 

260.8 

910.0 

831.5 

78.5 

.4252 

450.3 

7.21 

.1387 

60 

292.5 

1171.2 

261.9 

909.3 

830.7 

78.6 

.4267 

443.4 

7.10 

.1409 

61 

293.6 

1171.5 

263.0 

90S.5 

829.8 

78'.7 

.4281 

436.5 

6.99 

.1431 

62 

294.7 

1171.8 

264.1 

907.7 

828.9 

78.8 

.4295 

429.7 

6.88 

.1453 

63 

295.7 

1172.1 

265.2 

906.9 

828.0 

78.9 

.4309 

423.4 

6.78 

.1475 

64 

296.7 

1172.4 

266.2 

906.2 

827.3 

78.9 

.4323 

417.2 

6.68 

.1497 

65 

297.8 

1172.7 

267.2 

905.5 

826.5 

79.0 

.4337 

410.9 

6.58 

.1519 

66 

298.8 

1173.0 

268.3 

904.7 

825.6 

79.1 

.4350 

405.3 

6.49 

.1541 

67 

299.8 

1173.3 

269.3 

904,0 

824.8 

79.2 

.4363 

399.7 

6.40 

.1562 

68 

300.8 

1173.6 

270.3 

903.3 

824.1 

79.2 

.4376 

394.1 

6.31 

.1584 

69 

301.7 

1173.9 

271.2 

902.7 

823.4 

79.3 

.4389 

•389.1 

6.23 

.1606 

70 

302.7 

1174.3 

272.2 

902.1 

822.7 

79.4 

.4402 

383.4 

6.14 

.1628 

71 

303.7 

1174.6 

273.2 

901.4 

821.9 

79.5 

.4415 

378.4 

6.06 

.1649 

72 

304.6 

1174.9 

274.1 

900.8 

821.3 

79.5 

.4428 

373.4 

5.98 

.1671 

73 

305.5 

1175.2 

275.1 

900.1 

820.5 

79.6 

.4440 

369.1 

5.91 

.1693 

74 

306.5 

1175.4 

276.0 

899.4 

819.7 

79.7 

.4452 

364.1 

5.83 

.1714 

75 

307.4 

1175.7 

276.9 

898.8 

819.1 

79.7 

.4464 

359.7 

5.76 

.1736 

76 

308.3 

1176.0 

277.8 

898.2 

818.4 

79.8 

.4476 

355.3 

5.69 

.1757 

77 

309.2 

1176.2 

278.7 

897.5 

817.6 

79.9 

.4487 

351.0 

5.62 

.1779 

78 

310.1 

1176.5 

279.6 

896.9 

817.0 

79.9 

.4499 

346.6 

5.55 

.1801 

79 

310.9 

1176.8 

280.5 

896.3 

816.3 

80.0 

.4511 

342.8 

5.49 

.1822 

80 

311.8 

1177.0 

281.4 

895.6 

815.5 

80.1 

.4522 

338.5 

5.42 

.1843 

81 

312.7 

1177.3 

282.3 

895.0 

814.9 

80.1 

.4534 

334.7 

5.36 

.1865 

82 

313.5 

1177.6 

283.2 

894.4 

814.2 

80.2 

.4545 

331.0 

5.30 

.1886 

83 

314.4 

1177.8 

284.1 

893.7 

813T.4 

80.3 

.4557 

327.2 

5.24 

.1908 

84 

315.2 

1178.1 

285.0 

893.1 

812.8 

80.3 

.4568 

323.5 

5.18 

.1930 

85 

316.0 

1178.3 

285.8 

892.5 

812.1 

80.4 

.4579 

319.7 

5.12 

.1951 

86 

316.8 

1178.6 

286.7 

891.9 

811.5 

80.4 

.4590 

316.6 

5.07 

.1973 

87 

317.6 

1178.8 

287.5 

891.3 

810.8 

80.5 

.4601 

312.9 

5.01 

.1994 

88 

318.4 

1179.1 

288.4 

890.7 

810.2 

80.5 

.4612 

309.8 

4.96 

.2016 

89 

319.2 

1179.3 

289.2 

890.1 

809.5 

80.6 

.4622 

306.6 

4.91 

.2037 

90 

320.0 

1179.6 

290.0 

889.6 

808.9 

80.7 

.4633 

303.5 

4.86 

.2058 

91 

320.8 

1179.8 

290.8 

889.0 

808.3 

80.7 

.4643 

300.4 

4.81 

.2080 

92 

321.6 

1180.0 

291.6 

888.4 

807.6 

80.8 

.4653 

297.3 

4.76 

.2101 

93 

322.4 

1180.3 

292.4 

887.9 

807.1 

80.8 

.4663 

294.1 

4.71 

.2122 

94 

323.1 

1180.5 

293.2 

887.3 

806.4 

80.9 

.4673 

291.0 

4.66 

.2144 

95 

323.9 

1180.7 

294.0 

886.7 

805.8 

80.9 

.4683 

288.5 

4.62 

.2165 

96 

324.6 

1181.0 

294.8 

886.2 

805.2 

81.0 

.4693 

285.4 

4.57 

.2186 

146 


SATURATED   STEAM 


Properties  of  Saturated  Steam 


Abso- 
lute   Tem- 
Press.  peratur* 
Lbs.     Deg. 
per      Fah. 
Sq.  in. 

Heat 

Volume 
Relative.    Specific.  Density. 
Water        Cu.  ft.      Wt.  of 
at             per  Ib.     1  cu.  ft 
39°Ft                      Lb.. 

Total 

of 
Liquid 

of 
Vapor- 
ization 

of 
Internal 
Work 

of          Entropy 

External         of 
Work      Liquid 

97 

325.4 

1181.2 

295.6 

885.6 

804.6 

81.0 

.4703 

282.9 

4.53 

.2208 

98 

326.1 

1181.4 

296.4 

885.0 

803.9 

81.1 

.4713 

280.4 

4.49 

9  •>•>() 

99 

326.9 

1181.6 

297.1 

884.5 

803.4 

81.1 

.4723 

277.3 

4.44 

.'2250 

100 

327.6 

1181.9 

297.9 

884.0 

802.8 

81.2 

.4733 

274.8 

4.40 

.2271 

101 

328.3 

1182.1 

298.6 

883.5 

802.3 

81.2 

.4743 

272.3 

4.36 

.2293 

102 

329.0 

1182.3 

299.4 

882.9 

801.6 

81.3 

.4753 

269.8 

4.32 

.2314 

103 

329.7 

1182.5 

300.1 

882.4 

801.1 

81.3 

.4762 

267.3 

4.28 

.2335 

104 

330.4 

1182.7 

300.9 

881.8 

800.4 

81.4 

.4771 

264.8 

4.24 

.2356 

105 

331.1 

1182.9 

301.6 

881.3 

799.9 

81.4 

.4780 

262.9 

4.21 

.2378 

106 

331.8 

1183.1 

302.3 

880.8 

799.3 

81.5 

.4790 

260.4 

4.17 

.2399 

107 

332.5 

1183.4 

303.0 

880.4 

798.9 

81.5 

.4799 

257.9 

4.13 

.2420 

108 

333.2 

1183.6 

303.8 

879.8 

798.2 

81.6 

.4808 

256.1 

4.10 

.2441 

109 

333.9 

1183.8 

304.5 

879.3 

797.7 

81.6 

.4817 

253.6 

4.06 

.2462 

110 

334.6 

1184.0 

305.2 

878.8 

797.1 

81.7 

.4826 

251.7 

4.03 

.2484 

111 

335.2 

1184.2 

305.9 

878.3 

796.6 

81.7 

.4835 

249.2 

3.99 

.2505 

112 

335.9 

1184.4 

306.6 

877.8 

796.1 

81.7 

.48"43 

247.3 

3.96 

.2526 

113 

336.5 

1184.6 

307.3 

877.3 

795.5 

81.8 

.4852 

245.4 

3.93 

.254^ 

114 

337.2 

1184.8 

308.0 

876.8 

795.0 

81.8 

.4860 

242.9 

3.89 

.2568 

115 

337.9 

1185.0 

308.7 

876.3 

794.4 

81.9 

.4869 

241.1 

3.86 

.2589 

116 

338.5 

1185.2 

309.4 

875.8 

793.9 

81.9 

.4877 

239.2 

3.83 

.2610 

117 

339.1 

1185.4 

310.0 

875.4 

793.5 

81.9 

.4886 

237.3 

3.80 

.2631 

118 

339.8 

1185.6 

310.7 

874.9 

792.9 

82.0 

.4894 

235.4 

3.77 

.2653 

119 

340.4 

1185.8 

311.4 

874.4 

792.4 

82.0 

.4903 

233.6 

3.74 

.2674 

120 

341.0 

1186.0 

312.0 

874.0 

791.9 

82.1 

.4911 

231.7 

3.71 

.2695 

121 

341.7 

1186.2 

312.7 

873.5 

791.4 

82.1 

.4919 

229.8 

3.68 

.2715 

122 

342.3 

1186.3 

313.3 

873.0 

790.8 

82.2 

.4927 

227.9 

3.65 

.2736 

123 

342.9 

1186.5 

314.0 

872.5 

790.3 

82.2 

.4935 

226.7 

3.63 

.2757 

124 

343.5 

1186.7 

314.6 

872.1 

789.9 

82.2 

.4943 

224.8 

3.60 

.2779 

125 

344.1 

1186.9 

315.2 

871.7 

789.4 

82.3 

.4951 

223.0 

3.57 

.2800 

126 

344.7 

1187.1 

315.9 

871.2 

788.9 

82.3 

.4959 

221.7 

3.55 

.2820 

127 

345.3 

1187.3 

316.5 

870.8 

788.4 

82.4 

.4967 

219.8 

3.52 

.2841 

128 

345.9 

1187.4 

317.1 

870.3 

787.9 

82.4 

.4974 

218.0 

3.49 

.2862 

129 

346.5 

1187.6 

317.7 

869.9 

787.5 

82.4 

.4982 

216.7 

3.47 

.2883 

130 

347.1 

1187.8 

318.4 

869.4 

786.9 

82.5 

.4990 

214.8 

3.44 

.2904 

131 

347.7 

1188.0 

319.0 

869.0 

786.5 

82.5 

.4997 

213.6 

3.42 

.2925 

132 

348.3 

1188.2 

319.6 

868.6 

786.1 

82.5 

.5005 

211.7 

3.39 

.2946 

133 

348.9 

1188.4 

320.2 

868.2 

785.6 

82.6 

.5012 

210.5 

3.37 

.2967 

134 

349.5 

1188.5 

320.8 

867.7 

785.1 

82.6 

.5020 

209.2 

3.35 

.2988    - 

135 

350.0 

1188.7 

321.4 

867.3 

784.7 

82.6 

.5027 

207.3 

3.32 

.3009 

136 

350.6 

1188.9 

322.0 

866.9 

784.2 

82.7 

.5035 

206.1 

3.30 

.3030 

SAT CRATED   STEAM 


147 


Properties  of  Saturated  Steam 


Abso- 
ute     Tem- 

Heat 

Volume 

3ress 

perature 

of 

of 

of 

of 

Entropy 

Relative. 

Specific. 

Density. 

Lbs. 
per 

Deg. 
Fah. 

Total 

Liquid 

Vapor- 

ization 

Internal 
Work 

External 
Work 

of 
Liquid 

Water 
at 

Cu.  ft. 
per  Ib. 

Wt.  of 
1  cu.  ft. 

Sq.  in 

39°F. 

Lb«. 

137 

351.2 

1189.0 

322.6 

866.4 

783.7 

82.7 

.5042 

204.8 

3.28 

.3051 

38 

351.7 

1189.2 

323.2 

866.0 

783.3 

82.7 

.5049 

203.0 

3.25 

.3072 

139 

352.3 

1189.4 

323.8 

865.6 

782.8 

82.8 

.5056 

201.7 

3.23 

.3092 

140 

352.9 

1189.5 

324.4 

865.1 

782.3 

82.8 

.5064 

200.5 

3.21 

.3113 

141 

353.4 

1189.7 

325.0 

864.7 

781.9 

82.8 

.5071 

199.2 

3.19 

.3134 

142 

353.9 

1189.9 

325.6 

864.3 

781.4 

82.9 

.5078 

198.0 

3.17 

.315& 

4:j 

354.5 

1190.1 

326.1 

864.0 

781.1 

82.9 

.5085 

196.7 

3.15 

.3176 

144 

355.0 

1190.2 

326.7 

863.5 

780.6 

82.9 

.5092 

195.5 

3.13 

.3197 

.45 

355.6 

1190.4 

327.2 

863.2 

780.2 

83.0 

.5099 

194.2 

3.11 

.3218 

,4«j 

356.1 

1190.6 

327.8 

862.8 

779.8 

83.0 

.5106 

193.0 

3.09 

.3239- 

.47 

356.7 

1190.7 

328.3 

862.4 

779.4 

83.0 

.5113 

191.7 

3.07 

.3259- 

.48 

357.2 

1190.9 

328.9 

862.0 

778.9 

83.1 

.5119 

190.5 

3.05 

.3280- 

'.4!» 

357.7 

1191.0 

329.4 

861.6 

778.5 

83.1 

.5126 

189.2 

3.03 

.3300 

:50 

358.3 

1191.2 

330.0 

861.2 

778.1 

83.1 

.5133 

188.0 

3.01 

.3321 

151 

358.8 

1191.4 

330.5 

860.9 

777.7 

83.2 

.5140 

186.7 

2.99 

.3342: 

L52 

359.3 

1191.5 

331.1 

860.4 

777.2 

83.2 

.5146 

185.5 

2.97 

.336a 

5:j 

359.8 

1191.7 

331.6 

860.1 

776.9 

83.2 

.5153 

184.2 

2.95 

.3384 

360.3 

1191.8 

332.2 

859.6 

776.3 

83.3 

.5160 

183.6 

2.94 

.3405- 

360.9 

1192.0 

332.7 

859.3 

776.0 

83.3 

.5166 

182.3 

2.92 

.342$ 

i5t; 

361.4 

1192.2 

333.3 

858.9 

775.6 

83.3 

.5173 

181.1 

2.90 

.3447 

57 

361.9 

1192.3 

333.8 

858.5 

775.2 

83.3 

.5179 

179.8 

2.88 

.3467 

362.4 

1192.5 

334.3 

858.2 

774.8 

83.4 

.5186 

179.2 

2.87 

.3488 

59 

362.9 

1192.7 

334.9 

857.8 

774.4 

83.4 

.5192 

178.0 

2.85 

.3509* 

60 

363.4 

1192.8 

335.4 

857.4 

774.0 

83.4 

.5198 

176.7 

2.83 

.3530 

fil 

363.9 

1193.0 

335.9 

857.1 

773.7 

83.4 

.5205 

176.1 

2.82 

.3551 

.62 

364.4 

1193.1 

336.4 

856.7 

773.2 

83.5 

.5211 

174.9 

2.80 

.357± 

63 

364.9 

1193.3 

337.0 

856.3 

772.8 

83.5 

.5217 

173.6 

2.78 

.3593 

64 

365.4 

1193.4 

337.5 

855.9 

772.4 

83.5 

.5224 

173.0 

2.77 

.3614 

65 

365.9 

1193.6 

338.0 

855.6 

772.0 

83.6 

.5230 

171.7 

2.75 

.3635 

66 

366.4 

1193.7 

338.5 

855.2 

771.6 

83.6 

.5236 

171.1 

2.74 

.3655 

67 

366.8 

1193.9 

339.0 

854.9 

771.3 

83.6 

.5242 

169.9 

2.72 

.3675- 

68 

367.3 

1194.0 

339.5 

854.5 

770.9 

83.6 

.5248 

169.2 

2.71 

.3695 

69 

367.8 

1194.2 

340.0 

854.2 

770.5 

83.7 

.5254 

168.0 

2.69 

.3716 

70 

368.3 

1194.3 

340.5 

853.8 

770.1 

83.7 

.5260 

167.4 

2.68 

.3737 

71 

368.8 

1194.4 

341.0 

853.4 

769.7 

83.7 

.5266 

166.1 

2.66 

.3758 

72 

369.2 

1194.6 

341.5 

853.1 

769.4 

83.7 

.5272 

165.5 

2.65 

.3778 

73 

369.7 

1194.7 

342.0 

852.7 

768.9 

83.8 

.5278 

164.3 

2.63 

.3799 

74 

370.2 

1194.8 

342.5 

852.3 

768.5 

83.8 

.5284 

163.6 

2.62 

.3820 

75 

370.6 

1195.0 

343.0 

852.0 

768.2 

83.8 

.5290 

162.4 

2.60 

.3841 

76 

371.1 

1195.1 

343.5 

851.6 

767.8 

83.8 

.5296 

161.8 

2.59 

.3862 

148 


SATURATED   STEAM 


Properties  of  Saturated  Steam 


Abso- 

lute 

Tem- 

Heat 

Volume 

Press. 
Lbs. 

perature 
Deg. 

Total 

of 
Liquid 

of 
Vapor- 

of 
Internal 

of 
External 

Entropy 
of 

Relative. 
Water 

Specific. 
Cu.  ft. 

Density. 
Wt.  of 

per 

Fah. 

ization 

Work 

Work 

Liquid 

at 

per  Ib. 

1  cu.  ft. 

Sq.in 

39°F. 

Lbs. 

177 

371.6 

1195.3 

344.0 

851.3 

767.5 

83.8 

.5302 

160.5 

2.57 

.3883 

178 

372.0 

1195.4 

344.4 

851.0 

767.1 

83.9 

.5308 

159.9 

2.56 

.3904 

179 

372.5 

1195.6 

344.9 

850.7 

766.8 

83.9 

.5313 

159.3 

2.55 

.3925 

180 

373.0 

1195.7 

345.4 

850.3 

766.4 

83.9 

.5319 

158.0 

2.53 

.3945 

181 

373.4 

1195.9 

345.9 

850.0 

766.1 

83.9 

.5325 

157.4 

2.52 

.3966 

182 

373.9 

1196.0 

346.4 

849.6 

765.6 

84.0 

.5331 

156.7 

2.51 

.3987 

183 

374.3 

1196.1 

346.8 

849.3 

765.3 

84.0 

.5336 

155.5 

2.49 

.4008 

184 

374.8 

1196.2 

347.3 

848.9 

764.9 

84.0 

.5342 

154.9 

2.48 

.4029 

185 

375.2 

1196.4 

347.8 

848.6 

764.6 

84.0 

.5347 

154.3 

2.47 

.4049 

186 

375.7 

1196.5 

348.2 

848.3 

764.3 

84.0 

.5353 

153.6 

2.46 

.4070 

187 

376.1 

1196.6 

348.7 

847.9 

763.8 

84.1 

.5359 

152.4 

2.44 

.4090 

188 

376.6 

1196.8 

349.2 

847.6 

763.5 

84.1 

.5364 

151.8 

2.43 

.4111 

189 

377.0 

1196.9 

349.6 

847.3 

763.2 

84.1 

.5370 

151.1 

2.42 

.4132 

190 

377.4 

1197.1 

350.1 

847.0 

762.9 

84.1 

.5375 

150.5 

2.41 

.4153 

191 

377.9 

1197.2 

350.5 

846.7 

762.5 

84.2 

.5381 

149.9 

2.40 

.4174 

192 

378.3 

1197.3 

351.0 

846.3 

762.1 

84.2 

.5386 

148.7 

2.38 

.4194 

193 

378.7 

1197.4 

351.4 

846.0 

761.8 

84.2 

.5391 

148.0 

2.37 

.4215 

194 

379.2 

1197.6 

351.9 

845.7 

761.5 

84.2 

.5397 

147.4 

2.36 

.4236 

195 

379.6 

1197.7 

352.4 

845.3 

761.1 

84.2 

.5402 

146.8 

2.35 

.4257 

196 

380.0 

1197.8 

352.8 

845.0 

760.8 

84.2 

.5408 

146.1 

2.34 

.4278 

197 

380.5 

1198.0 

353.3 

844.7 

760.4 

84.3 

.5413 

144.9 

2.32 

.4298 

198 

380.9 

1198.1 

353.7 

844.4 

760.1 

84.3 

.5418 

144.3 

2.31 

.4318 

199 

381.3 

1198.2 

354.1 

844.1 

759.8 

84.3 

.5423 

143.6 

2.30 

.4338 

200 

381.7 

1198.4 

354.6 

843.8 

759.5 

84.3 

.5429 

143.0 

2.29 

.4359 

201 

382.1 

1198.5 

355.0 

843.5 

759.1 

84.4 

.5434 

142.4 

2.28 

.4379 

202 

382.6 

1198.6 

355.4 

843.2 

758.8 

84.4 

.5439 

142.8 

2  27 

.4399 

203 

383.0 

1198.8 

355.9 

842.9 

758.5 

84.4 

.5444 

142.2 

2^26 

.4420^ 

204 

383.4 

1198.9 

356.3 

842.6 

758.2 

84.4 

.5449 

140.5 

2  25 

.4441^ 

205 

383.8 

1199.0 

356.8 

842.2 

757.8 

84.4 

.5454 

139.9 

2.24 

.4461 

206 

384.2 

1199.1 

357.2 

841.9 

757.4 

84.5 

.5459 

139.3 

2.23 

.4482 

207 

384.6 

1199.3 

357.6 

841.7 

757.2 

84.5 

.5465 

138.6 

2  22 

.4503 

208 

385.0 

1199.4 

358.0 

841.4 

756.9 

84.5 

.5470 

138.0 

2i21 

.4524 

209 

385.5 

1199.5 

358.5 

841.0 

756.5 

84.5 

.5475 

137.4 

2.20 

.454 

210 

385.9 

1199.6 

358.9 

840.7 

756.2 

84.5 

.5480 

136.8 

2.19 

.456o 

211 

386.3 

1199.8 

359.3 

840.5 

756.0 

84.5 

.5485 

136.1 

2.18 

.458 

212 

386.7 

1199.9 

359.7 

840.2 

755.6 

84.6 

.5489 

135.5 

2.17 

.460 

213 

387.1 

1200.0 

360.1 

839.9 

755.3 

84.6 

.5494 

134.9 

2.16 

.462 

214 

387.5 

1200.1 

360.6 

839.5 

754.9 

84.6 

.5499 

134.3 

2.15 

.464 

215 

387.9 

1200.2 

361.0 

839.2 

754.6 

84.6 

.5504 

133.6 

2.14 

.466 

216 

388.3 

1200.4 

361.4 

839.0 

754.4 

84.6 

.5509 

133.0 

2.13 

.469 

217 

388.7 

1200.5 

361.8 

838.7 

754.1 

84.6 

.5514 

132.4 

2.12 

.4711 

21  S 

389.1 

1200.6 

362.2 

838.4 

753.8 

84.6 

.5519 

131.8 

2.11 

.4731 

SATURATED   STEAM 


149 


Properties  of  Saturated  Steam 


Abso- 
lute 

Tem- 

^ 

(eat 

Vo 

urne 

Pr^ss. 
Lbs. 
per 
Sq.  in 

perature 
Deg. 
Fah. 

Total 

of 

Liquid 

of 

Vapor- 
ization 

of 
Internal 
Work 

of 
External 
Work 

Entropy 
of 
Liquid 

Relative. 
Water 

39a°V 

Specific. 
Cu.  ft. 
per  Ib. 

Density. 
Wt.  of 
1  cu.  ft. 
Lbs. 

219 

389.4 

1200.7 

362.6 

838.1 

753.4 

84.7 

.5524 

131.1 

2.10 

.4751 

220 

389.8 

1200.8 

363.0 

837.8 

753.1 

84.7 

.5529 

131.1 

2.10 

.4772 

221 

390.2 

1201.0 

363.5 

837.5 

752.8 

84.7 

.5533 

130.5 

2.09 

.4792 

'222 

390.6 

1201.1 

363.9 

837.2 

752.5 

84.7 

.5538 

129.9 

2.08 

.4813 

223 

391.0 

1201.2 

364.3 

836.9 

752.2 

84.7 

.5543 

129.3 

2.07 

.4834 

224 

391.4 

1201.3 

364.7 

836.6 

751.9 

84.7 

.5548 

128.6 

2.06 

.4855 

225 

391.8 

1201.4 

365.1 

836.3 

751.6 

84.7 

.5553 

128.0 

2.05 

.4876 

:226 

392.2 

1201.6 

365.5 

836.1 

751.3 

84.8 

.5557 

127.4 

2.04 

.4896 

227 

392.5 

1201.7 

365.9 

835.8 

751.0 

84.8 

.5562 

126.8 

2.03 

.4917 

228 

392.9 

1201.8 

366.3 

835.5 

750.7 

84.8 

.5567 

126.2 

2.02 

.4939 

229 

393.3 

1201.9 

366.7 

835.2 

750.4 

84.8 

.5571 

126.2 

2.02 

.4959 

230 

393.7 

1202.0 

367.1 

834.9 

750.1 

84.8 

.5576 

125.5 

2.01 

.4979 

231 

394.1 

1202.1 

367.5 

834.6 

749.8 

84.8 

.5581 

124.9 

2.00 

.5000 

L':J2 

394.4 

1202.2 

367.9 

834.3 

749.5 

84.8 

.5585 

124.3 

1.99 

.5021 

2!i;! 

394.8 

1202.4 

368.3 

834.1 

749.2 

84.9 

.5590 

123.7 

1.98 

.5041 

2:;  4 

395.2 

1202.5 

368.6 

833.9 

749.0 

84.9 

.5594 

123.7 

1.98 

.5062 

235 

395.6 

1202.6 

369.0 

833.6 

748.7 

84.9 

.5599 

123.0 

1.97 

.5082 

2;;t; 

395.9 

1202.7 

369.4 

833.3 

748.4 

84.9 

.5603 

122.4 

1.96 

.5103 

237 

396.3 

1202.8 

369.8 

833.0 

748.1 

84.9 

.5608 

121.8 

1.95 

.5123 

:  L'i'.N 

396.7 

1202.9 

370.2 

832.7 

747.8 

84.9 

.5612 

121.2 

1.94 

.5144 

23!t 

397.0 

1203.0 

370.6 

832.4 

747.5 

84.9 

.5617 

121.2 

1.94 

.5165 

240 

397.4 

1203.2 

371.0 

832.2 

747.3 

84.9 

.5621 

120.5 

1.93 

.5186 

241 

397.8 

1203.3 

371.3 

832.0 

747.0 

85.0 

.5626 

119.9 

1.92 

.5206 

24_ 

398.1 

1203.4 

371.7 

831.7 

746.7 

85.0 

.5630 

119.3 

1.91 

.5226 

24.", 

398.5 

1203.5 

372.1 

831.4 

746.4 

85.0 

.5635 

119.3 

1.91 

.5247 

244 

398.8 

1203.6 

372.5 

831.1 

746.1 

85.0 

.5639 

118.7 

1.90 

.5268 

-  5 

399.2 

1203.7 

372.8 

830.9 

745.9 

85.0 

.5643 

118.0 

1.89 

.5289 

•2  i; 

399.6 

1203.8 

373.2 

830.6 

745.6 

85.0 

.5648 

117.4 

1.88 

.5311 

L,  7 

399.9 

1203.9 

373.6 

830.3 

745.3 

85.0 

.5652 

116.8 

1.87 

.5332 

24  N 

400.3 

1204.0 

374.0 

830.0 

745.0 

85.0 

.5656 

116.8 

1.87 

.5353 

2   !< 

400.6 

1204.1 

374.3 

829.8 

744.8 

85.0 

.5661 

116.2 

1.86 

.5373 

250 

401.0 

1204.2 

374.7 

829.5 

744.5 

85.0 

.5665 

115.5 

1.85 

.5393 

260 

404.5 

1205.3 

378.4 

826.9 

741.7 

85.2 

.5707 

111.2 

1.78 

.5601 

270 

407.8 

1206.3 

381.9 

824.4 

739.2 

85.2 

.5748 

107.4 

1.72 

.5809 

280 

411.1 

1207.3 

385.3 

822.0 

736.7 

85.3 

.5787 

103.7 

1.66 

.602 

290 

414.3 

1208.3 

388.6 

819.7 

734.3 

85.4 

.5826 

100.5 

1.61 

.622 

300 

417.4 

1209.3 

391.9 

817.4 

732.0 

85.4 

.5863 

96.8 

1.55 

.644 

310 

420.4 

1210.2 

395.0 

815.2 

729.8 

85.4 

.5898 

94.3 

1.51 

.664 

t325 

424.8 

1211.5 

399.6 

811.9 

726.4 

85.5 

.5950 

89.9 

1.44 

.696 

350 

431.8 

1213.1 

406.7 

806.4 

721.1 

85.3 

.6028 

83.7 

1.34 

.749 

375 

438.5 

1215.5 

413.7 

801.7 

716.4 

85.2 

.6107 

78.1 

1.25 

.802 

400 

444.8 

1217.6 

420.4 

797.3 

712.1 

85.2 

.6181 

73.1 

1.17 

.855 

150 


VAPORS  OF  AMMONIA  AND  SULPHUR  DIOXIDE 


Saturated  Vapor  of  Ammonia. 


Heat 

Temp. 

Pressure 

of 

of 

of 

Volume 

Density 

Deg. 
Fabr. 

Lbs.  per 
sq.  in. 

Total 

in 
Liquid 

Vapor- 
ization 

Internal 
Work 

External 
Work 

Entropy 

Cu.  ft. 
per  Ib. 

Lbs.  per 
cu.  ft. 

—25 

15.43 

530.9 

—62.1 

593.0 

543.2 

49.8 

—.133 

17.1 

.0585 

—15 

20.48 

537.7 

—51.5 

589.2 

538.5 

50.7 

—.110 

13.5 

.0741 

—  5 

26.41 

544.0 

—40.7 

584.7 

533.2 

51.5 

—.085 

10.5 

.0952 

5 

33.71 

549.0 

—29.9 

578.9 

526.7 

52.2 

—.062 

8.40 

.119 

15 

42.57 

553.0 

—18.9 

571.9 

519.2 

52.7 

—.039 

6.72 

.149 

25 

53.35 

555.7 

—  7.9 

563,6 

510.4 

53.2 

—.016 

5.46 

.183 

30 

59.44 

556.8 

.  —  2.3 

559.1 

505.6 

53.5 

—.005 

4.91 

.204 

32 

61.87 

557.3 

0.0 

557.3 

503.6 

53.7 

.000 

4.69 

.213 

40 

72.91 

558.2 

11.1 

547.1 

493.2 

53.9 

.022 

4.02 

.249 

50. 

89.18 

558.5 

20.5 

538.0 

484.0 

54.0 

.041 

3.30 

.303 

60 

108.08 

558.0 

32.1 

525.9 

471.7 

54.2 

.069 

2.74 

.365 

70 

129.83 

556.4 

43.5 

512.9 

458.6 

54.3 

.084 

2.29 

.437 

80 

154.73 

554.3 

55.4 

498.9 

444.8 

54.1 

.107 

1.83 

.546 

90 

182.82 

551.3 

67.4 

483.9 

430.0 

53.9 

.128 

1.63 

.613 

100 

214.45 

548.0 

79.6 

468.4 

414.8 

53.6 

.150 

1.39 

.719 

Saturated 

Vapor  of 

Sulphur  Dioxide. 

Heat 

Temp. 
Deg. 
Fahr. 

Pressure 
Lbs.  per 
sq.  in. 

Total 

in 
Liquid 

of 

Vapor- 
ization 

of 

Internal 
Work 

of 

External 
Work 

Entropy 

Volume 
Cu.  ft. 
per  Ib. 

Density 
Lbs.  per 
cu.ft. 

—25 
—15 
—  5 

4.98 
6.82 
8.96 

159.4 
160.4 
161.5 

—17.1 
—14.4 
—11.3 

176.5 

174.8 
172.8 

163.1 
161.1 
158.8 

13.4 
13.7 
14.0 

—.0370 
—.0306 
—.0241 

14.5 
10.8 

8.40 

.0690 
.0926 
.119 

5 

11.81 

162.5 

—  8.2 

170.7 

156.4 

14.3 

—.0176 

6.67 

.150 

15 

15.22 

163.1 

—  5.1 

168.2 

153.7 

14.5 

—.0111 

5.15 

.194 

25 

19.21 

163.5 

—  2.0 

165.5 

150.9 

14.6 

—.0046 

4.12 

.243 

30 

21.61 

163.6 

—  0.5 

164.1 

149.4 

14.7 

—.001:; 

3.70 

.270 

32 
40 

22.48 
26.90 

163.6 
163.7 

0.0 

2.7 

163.6 
161.0 

148.8 
146.1 

14.8 
14.9 

.0000 
.0052 

3.56 
3.00 

.281 
.333 

50 

33.30 

163.6 

6.0 

157.6 

142.7 

14.9 

.0117 

2.43 

.412 

60 

40  83 

163.4 

9.4 

154.0 

139.1 

14.9 

.0182 

1.99 

.503 

70 

49.65 

162.9 

12.8 

150.1 

135.2 

14.9 

.0247 

1.64 

.611 

80 

59.74 

162.2 

16.2 

146.0 

131.1 

14.9 

.0312 

1.35 

.739 

90 

71  41 

16l!2 

19.8 

141.4 

126.6 

14.8 

.0377 

1.13 

.888 

100 

84.63 

160.2 

23.4 

136.8 

122.2 

14.6 

.0442 

.943 

1.06 

MAGNETIC   CONSTANTS  151 


Magnetic  Properties  of  Iron  and  Steel 

H  =  magnetizing  force  =  1.258  ampere  turns  per  cm.=  .495  ampere 
turns  per  inch. 

B  =  kilo-lines  per  square  centimeter. 
B"=  kilo-lines    per    square    inch. 
A  —  ampere   turns   per  centimeter  length. 
A"=  ampere   turns    per   inch   length. 


Permeability  /A  =  B-=-H. 
(Values    given    by    Sheldon    and    Foster    for    avei'age 
metal.) 

first    quality 

H 

A 

A" 

Sheet  Metal 
B           B" 

Cast  Iron 
B        B" 

Wrought  Iron 
B            B" 

CaSt  Steel 
B           B1 

10 
20 
30 
40 

7.95 

15.90 
23.85 
31.80 

20.2 
40.4 
60.6 
80.8 

14.3 
15.6 
16.2 
16.6 

92.2 

100.7 
104.5 
107.1 

4.3 
5.7 
6.5 
7.1 

27.7 
36.8 
41.9 
45.8 

13.0 
14.7 
15.3 
15.7 

83. 
94. 
98. 
101. 

8 
fr 
i 
2 

11.5 
13.8 
14.9 
15.5 

74.2 
89.0 
96.1 
100.0 

50 
60 

70 

39.75 

47.70 
55.65 

101.0 
121.2 
141.4 

16.9 
17.3 
17.5 

109.0 
111.6 
112.9 

7.6 
8.0 
8.4 

49.0 
51.6 
53.2 

16.0 
16.3 
16.5 

103. 

105. 
106. 

2 
2 
5 

16.0 
16.5 
16.9 

103.2 
106.5 
109.0 

80 
90 
100 

63.65 
71.60 
79.50 

161.6 
181.8 
202.0 

17.7 
18.0 
18.2 

114.1 
116.1 
117.3 

8.7 
9.0 
9.4 

56.1 
58.0 
60.6 

16.7 
16.9 
17.2 

107. 
109. 
110. 

8 

0 
9 

17.2 
17.4 
17.7 

110.0 
112.2 
114.1 

150 
200 
250 

119.25 
159.0 
198.8 

303.0 
404.0 
505.0 

19.0 
19.6 
20.2 

122.7 
126.5 
130.2 

10.6 
11.7 
12.4 

68.3 

75.5 
80.0 

18.0 
18.7 
19.2 

116. 
120. 
123. 

1 
8 
9 

18.5 
19.2 
19.7 

119.2 
123.9 
127.1 

300 

400 
600 

238.5 
318.0 
477.0 

606.0 
808.0 
1212 

20.7 
21.0 
21.5 

133.5 

135.0 
138.0 

13.2 

85.1 

19.7 

127. 

1 

20.1 

129.6 

800 

1000 
1200 

637.0 
795 
954 

1616 
2020 
2424 

21.8 
22.0 
22.3 

140.0 
142.0 
144.0 

*   •.- 

Loss   Due  to   Hysteresis 

Loss  in  ergs  per  cubic  centimeter  is  L  =  *?  B  1<(!  in  which  77  is 
a  constant  depending  upon  the  iron. 

Values  of  tj: 


Very    soft    iron   wire  .002  Soft  annealed  cast  steel       .008 

Very  thin  soft  sheet  iron  .0015  Soft  machine  steel 

Thin   good   sheet   iron  .003  Cast    steel 

Thick   sheet   iron  .0033  Cast    iron 

Ordinary    sheet    iron  .004  Hardened  cast   steel  .025 

Transformer   cores  .003 


152  ELECTRICAL,  CONSTANTS 

Specific  Resistances  and  Temperature  Coefficients 


R=  r  (1  +  a  f)                              Ohms  per 
R  =  resistance  at  r°  C                      cir.  mil-foot 
r  =  resistance  at  o°  C                          at  0°  C 

Specific  Resistance           Temperature 
Microhms  per               Coefficient  a. 
cubic  centimeter          Divide  by  105 

Aluminum 

17.38 

2.889 

390 

Carbon,  graphite 

j      2400 
(    42000 

Carbon,  arc  light 

4000 

Constantine 

307 

51 

±  1 

Copper,  annealed 

9.44 

1.570 

388 

Copper,  hard 

9.64 

1.603 

German  silver 

125.0 

20.76 

28  to  44 

Gold 

2.07 

'  365 

la.   la.,  hard 

302 

50.2 

—1.1 

la.    la.,  soft 

283 

47.1 

0.5 

Iron 

58.0 

9.64 

453 

Iron,  wire 

57.2 

9.50 

Iron,  telegraph  wire 

90.0 

150 

Lead 

118 

19.6 

387 

Manganin 

287 

47.5 

±  1 

Mercury 

94.34 

88 

Nickel 

12.35 

500 

Nickeline  No.     I,  hard 

262 

43.6 

7.6 

Nickeline  No.     I,  soft 

245 

40.7 

7.7 

Nickeline  No.  II,  hard 

204 

33.9 

16.8 

Nickeline  No.  II,  soft 

194 

32.3 

18.1 

Platinum 

54.03 

8.98 

247 

Silver,  annealed 

1.49 

377 

Silver,   hard 

1.62 

Steel,  wire 

95 

15.8 

390 

Dilute  H  N  O3  30% 
H2  S  04  5% 
H2  S  O4  30% 

129  X  104 
486  X  104 
137  X  104 

[  Liquids 

H2  S  04  80% 

918  X  104 

fat  18°  C 

Zn  S  O4  24% 

214  X  105 

\ 

Water 

26.5X  108 

; 

V 

Megohms 

Benzine 

14  X  106 

Ebonite 

28  X  109 

Glass,  20°C 

91  X  1Q6 

Glass,  200°C 

22.7 

Gutta-percha,  24°C 

4.5  X  108 

Mica 

84  X  106 

Paraffine 

34  X  109 

Paraffine,  oil               sc 

8  X  106 

Shellac 

9  X  109 

Wood  Tar 

167  X  107 

ELECTRICAL    CONSTANTS 


153 


Specific  Inductive  Capacities. 


Specific 

Specific 

Medium. 

Inductive 

Medium. 

Inductive 

Capacity. 

Capacity. 

Air,  760  mm. 

1.0 

Mica 

6.64 

Alcohol 

26.0 

Paraffine,   solid 

1.96-2.3 

Beeswax 

1.8 

Paraffine,   oil 

1.92 

Ebonite 

2.2-3  2 

Petroleum 

2.05 

Glass,   light  flint 

6.72 

Shellac 

2.7-3.7 

Glass,   dense  flint 

7.38 

Sulphur 

2.8-3.9 

Glass,   hard  crown 

6.96 

Turpentine 

2  2 

Glass,    plate 

5.8-8.5 

Vacuum 

!999 

Gutta-percha 

2.5 

Water 

76.0 

Kerosene 

2-2.5 

Rate  of  Electrolytic  Deposition. 


Element. 

Mg.  per 
Coulomb. 

Element. 

Mg.  per 
Coulomb. 

Aluminum 
Copper 
Gold 
Hydrogen 
Lead 
Mercury 

.0935 
.3282 
.0679 
.0104 
.1072 
1.038 

Nickel 
Oxygen 
Platinum 
Silver 
Tin 
Zinc 

.3043 

.0831 
1.0095 
1.1186 
.6097 
.3365 

WIRES  AND  CABLES 

Fusing  of  Wires. 

:V 
I  =  ad  '  •    in  which   /  is  the  fusing  current  in  amperes,  d  is 

the  diameter  of  the  wire  and  a  is  a  constant,  given  as  follows  for 
various  materials  with  d  measured  either  in  inches,  centimeters 
or  millimeters. 

Constant  a. 


Material 

In. 

Cm. 

Mm. 

Alloy,    lead    (2)    tin    (1) 
Aluminum 

1318 
7585 

325.5 
1873 

10.3 
59.2 

Copper 
German  silver 

10244 
5230 

1530 
1292 

80.0 
40.8 

Iron 
Lead 

3148 
1379 

777.4 
340.6 

24.6 
10.8 

Platinum 
Tin 

5172 
1642 

1277 

405.5 

40.4 
12.8 

154 

WIRES    AND    CABLES 

Copper  Wire. 

To  obtain 

resistance  at 

30°C 

40°  C 

50°C 

60°C 

70CC 

80°C 

Multiply  values  in  it 

i  hie  by 

1.036 

1.072 

1.108 

1.144 

1.180 

1-216 

No. 
B.&S. 

Dia. 
inches 

Area 
Cir.  Mils 

Ohms  at  20QC 
1,000  ft-        Mile 

Wt.  Ibs. 
1,000  ft. 

Bare 
Mile 

3381.8 
2682.2 
2126.8 
1687.0 

Safe  Cap. 
W.  P. 

Amp. 
R.  C, 

210 

177 
150 
127 

0000 
000 
00 
0 

.460 
.4096 
.3648 
.3249 

211600 
167800 
133100 
105500 

.0489 
.0617 
.0778 
.0981 

.258 
.326 
.411 
.518 

640.5 
508.0 
402.8 
319.5 

312 
262 
L'20 
185 

1 
2 
3 

.2893 
.2576 
.2294 

83690 
66370 
52630 

.1237 
.1560 
.1967 

.653 

.824 
1.039 

253.3 
200.9 
159.3 

1337.4 

1060.8 
841.1 

156 
131 

110 

107 
90 
76 

4 
5 
6 

.2043 
.1819 
.1620 

41740 
33100 
26250 

.2480 
.3128 
.3944 

1.309 
1.651 

2.082 

126.4 
100.2 
79.46 

667.4 
529.6 
419.5 

92 

77 
65 

65 
54 
46 

7 
8 
9 

.1443 
.1285 
.1144 

20820 
16510 
13090 

.4973 
.6271 
.7908 

2.626 
3.311 
4.175 

63.02 
49.98 
39.63 

332.7 
263.9 
209.2 

46 

33 

10 
11 
12 

.1019 
.09074 
.08081 

10380 
8234 
6530 

.9972 
1.257 
1.586 

5.265 
6.640 
8.374 

31.43 

24.93 
19.77 

165.9 
131.6 
104.4 

23 

24 
17 

13 

14 
15 

.07196 
.06408 
.05707 

5178 
4107 
3257 

1.999 
2.521 
3.179 

10.555 
13.311 
16.785 

15.68 
12.43 
9.86 

82.9 
65.6 

52.0 

16 

12 

16 
17 
18 

.05082 
.04526 
.04030 

2583 

2048 
1624 

4.009 
5.055 
6.374 

21.168 
26.690 
33.655 

7.82 
6.20 
4.92 

41.3 
32.7 

26.0 

8 
5 

6 
3 

19 
20 
21 

.03589 
.03196 
.02846 

1288 
1022 
810 

8.038 
10.14 
12.78 

42.441 
53.539 
67.478 

3.90 
3.09 
2.45 

20.6 
16.3 
12.9 

22 
23 

24 

.02535 
.02257 
.02010 

642 
509 
404 

16.12 

20.32 
25.63 

1.95 
1.54 

1.22 

25 
26 
27 

.01790 
.01594 
.01420 

320 
254 
201 

32.31 

40.75 
51.38 

.97 
.769 
.610 

28 
29 
30 

.01264 
.01126 
.01003 

160 
127 

100 

64.79 
81.70 
103.0 

.484 
.384 
.304 

31 
32 
33 

.00893 
.00795 
.00708 

80 
63 
50 

129  9 
163.8 
206.6 

.241 
.191 
.152 

34 
35 
36 

.00631 
.00562 
.00500 

40 
32 
25 

260.5 
328.4 
414.2 

.120 
.096 
.076 

37 
38 
39 

40 

.00445 
.00397 
.00353 
.00315 

20 
16 
12 
10 

522.2 
658.5 
830.4 
1047.0 

.060 
.048 
.OS'S 
.030 

WIRES  AND  CABLES 

Aluminum  Wire. 


155 


To  obtain  resistance  at 
Multiply  values  in  ta* 

ble  by 

30°C 
1.036 

40°C 
1-072 

50°C 
1.109 

60°C 
1.145 

70°C    80°C 
1.181    1.217 

No. 
B.&S- 

Dia.  Bare 
inches 

Area 
Cir.  Mils 

Ohms  at  20QC 
1,000  ft.   Mile 

Wt.  Ibs. 
1,000  ft. 

Bare  Safe  Cap.  Amp. 
Mile    W.  P.  R.  C. 

0000 
000 
00 
0 

.460 
.4096 
.3648 
.3249 

211600 
167800 
133100 
105500 

.0804 
.1014 
.1278 
.1610 

.425 
.535 
.675 
.850 

192.0 
.152.3 
120.8 
95.9 

1014.0 
804.0 
637.7 
506.2 

262    177 
220    150 
185    127 
156    107 

1 

2 
3 

.2893 
.2576 
.2294 

83690 
66370 
52630 

.2032 
.2562 
.3230 

1.073 
1.353 
1.705 

76.0 
60.2 

47.8 

401.1 
318.0 
252.2 

131     90 
110     76 
92     65 

4 
5 
6 

.2043 

.1819 
.1620 

41740 
33100 
26250 

.4075 
.5140 
.6480 

2.152 
2.714 
3.421 

37.9 
.30.0 
23.8 

200.0 
158.6 
125.8 

77     54 
65     46 
55     39 

7 
8 
9 

.1443 

.1285 
.1144 

20820 
16510 
13090 

.8171 
1.050 
1.300 

4.314 
5.544 
6.864 

18.9 
15.0 
11.9 

99.8 
79.1 
62.7 

39     28 

10 
11 
12 

.1019 
.09074 
.08081 

10380 
8234 
6530 

1.640 

2.068 
2.605 

8.659 
10.92 
13.75 

9.42 
7.47 
5.93 

49.8 
39.4 
31.3 

27     20 
19     14 

13 
14 
15 

.07196 
.06408 
.05707 

5178 
4107 
3257 

3.284 
4.250 
5.111 

17.34 
22.44 
26.99 

4.71 
3.73 
2.96 

24.8 
19.7 
15.6 

13     10 

16 
17 
18 

.05082 
.04526 
.04030 

2583 
2048 
1624 

6.591 
8.315 
10.49 

34.80 
43.80 
59.39 

2.34 
1.86 
1.47 

12.4 
9.83 

7.78 

7     5 
4    2.5 

19 
20 
21 

.03589 
.03196 
.02846 

1288 
1022 
810 

13.60 
16.66 
21.03 

71.81 

87.96 

1.17 
.929 
.737 

6.18 
4.91 

22 
23 

24 

.02535 
.02257 
.02010 

642 
509 
404 

26.51 
33.42 
42.15 

.581 
.464 
.367 

25 
26 
27 

.01790 
.01594 
.01420 

320 

254 
201 

53.17 
67.05 
84.50 

.291 
.230 
.183 

28 
29 
30 

.01264 
.01126 
.01003 

160 
'  127 
100 

106.5 
134.4 
169.3 

.144 
.116 
.091 

31 
32 
33 

.00893 
.00795 
.00708 

80 
63 

50 

208.6 
269.5 
339.6 

.072 
.057 
.045 

34 
35 
36 

.00631 
.00562 
.00500 

40 
32 
25 

428.5 
540.0 
681.0 

.036 
.029 
.023 

37 
38 
39 

40 

.00445 
.00397 
.00353 

.00315 

20 
16 

io 

858.9 
1084 
1366 
1722 

.018 
.014 
.011 
.009 

156 


WIRES  AND  CABLES 

Copper  Cables. 


SizeB.&S.  Dia.  Bare 
orCir.  Mils   inches 

Ohms  at  20«C 
1.000  ft.    Mile 

Wt.  Ibs. 
1,000  ft- 

Bare 
Mile 

Safe  Cap.  Amp. 
W.  P.    R.  C. 

8 
6 
5 

.147 
.180 
.209 

.575 
.362 
.293 

3.040 
1.912 
1.550 

57 
85 
112 

301 
449 
591 

46 
65 

77 

33 
46 
54 

4 
3 
2 

.234 
.263 
.295 

.234 
.185 
.147 

1.236 

.980 
.776 

140 
178 
224 

739 
940 
1183 

92 
110 
131 

65 
76 

90 

1 
0 
00 

.325 

.378 
.425 

.129 
.0972 
.0772 

.681 
.513 

.407 

255 
338 
426 

1346 
1785 
2249 

156 
185 
220 

107 
127 

150 

000 
0000 
250000 

.475 
.524 
.568 

.0618 
.0478 
.0414 

.326 
.252 
.218 

532 

650 
790 

2809 
3432 
4171 

262 
.312 

350 

177 
210 
235 

300000 
350000 
400000 

.637 
.680 
.735 

.0340 
.0295 
.02,57 

.179 
.156 
.136 

949 
1092 
1224 

5010 
5766 
6463 

400 
450 
500 

270 
300 
330 

500000 
600000 
750000 

.820 
.900 
1.020 

.0204 
.0172 
.0138 

.108 
.0909 
.0727 

1550 
1874 
2331 

8184 
9895 
10208 

590 
680 
800 

390 
450 
525 

800000 
900000 
1000000 

1.037 
1.096 
1.157 

.0129 
.0115 
.0103 

.0683 
.0605 
.0543 

2462 

2815 
3138 

12999 
14863 
16579 

840 
920 
1000 

550 
600 
650 

1250000 
1500000 
2000003 

1.296 
1.412 
1.652 

.00828 
.00685 
.00517 

.0437 
.0362 
.0273 

3831 

4681 
6237 

20228 
24716 
32932 

1185 
1360 
1670 

750 
850 
1050 

Aluminum 

Cables. 

Cir. 

Mils 

Dia.  Bare 
inches 

Ohms  at  200C 
1,000  ft.    Mile 

Wt.  Ibs. 
1,000  ft. 

Bare 

Mile 

Safe  Cap.  Amp 
W.  P.   R.  C. 

250000 
300000 
350000 

.590 
.630 
.679 

.0668 
.0557 
.0477 

.353 
.294 
.252 

230 
276 
322 

1215 
1458 
1701 

294 
340 

378 

198 
228 
252 

400000 
450000 
500000 

.728 
.770 
.819 

.0417 
.0371 
.0334 

.220 
.196 
.176 

368 
414 

460 

1924 
2187 
2430 

420 
457 

495 

277 
302 
327 

550000 
600000 
650000 

.855 
.891 
.927 

.0304 
.0278 
.0257 

.161 
.147 
.136 

506 
552 

598 

2673 

2916 
3159 

532 

570 
604 

352 
378 
399 

700000 
750000 
800000 

.963 

.999 
1.035 

.0239 
.0223 
.0209 

.126 
.118 
.110 

644 
690 
736 

3402 
3645 

3888 

638 
672 
686 

420 
441 

462 

850000 
900000 
950000 
1000000 

1.062 
1.092 
1.125 
1.152 

.0196 
.0186 
.0176 
.0167 

.103 
.0982 
.0929 
.0882 

782 
828 
874 
920 

4131 
4374 

4617 
4860 

729 
772 
806 
840 

483 
504 
525 
546 

WIRES  AND  CABLES 

Cotton  Covered  Wire. 


157 


No. 
B.  &S. 

Dia.  Inches. 
Single         Double        Triple 

No. 
B.&S. 

Dia.  Inches. 
Single         Double 

Triple 

0000 

.480 

17 

.050 

.053 

.057 

000 

.430 

18 

.045 

.048 

.052 

00 

.385 

19 

.040 

.044 

.048 

0 

.339 

.343 

20 

.036 

.040 

.044 

1 

.303 

.307 

21 

.032 

.036 

.040 

2 

.272 

.276 

22 

.029 

.033 

.037 

3 

.242 

.247 

23 

.027 

.031 

.035 

4 

.211 

.216 

.220 

24 

.024 

.028 

.032 

5 

.189 

.194 

.198 

25 

.022 

.026 

.030 

6 

.169 

.174 

.178 

26 

.020 

.024 

7 

.151 

.156 

.160 

27 

.018 

.022 

8 

.136 

.141 

.145 

28 

.017 

.021 

9 

.121 

.126 

.130 

29 

.015 

.019 

10 

.108 

.112 

.116 

30 

.014 

.018 

11 

.097 

.101 

.105 

31 

.013 

.017 

12 

.087 

.091 

.095 

32 

.012 

.016 

13 

.078 

.082 

.086 

33 

.011 

.015 

14 

.070 

.074 

.078 

34 

.0103 

.0143 

15 

.063 

.067 

.071 

35 

.0096 

.0136 

16 

.056 

.059 

.063 

36 

.0085 

.0120 

The  B.  &  S.  Gauge 

The  diameter  of  any  wire  of  number  n  is 
_  0.3249 
"  1.123- 

and  the  area  in  circular  mils  an     =         °a 

1.261" 

These  equations  give  rise  to  the  following  approximate  ones 
for  slide  rule  calculation.  The  right  hand  side  of  the  equation 
gives  the  entire  logarithm,  but  only  the  decimal  part  is  to  be 
set  off  on  the  L  scale  on  the  back  of  the  slide  rule.  The  decimal 
point  will  probably  be  known  from  experience,  but  if  there  is 
any  doubt  it  may  be  determined  from  the  integral  part 
(characteristic)  of  the  logarithm  by  the  rules  on  page  4 
10  — n 
20 


log  dn 
log  a, 


(diameter,  inches) 


n  (area,  cir.  mils) 


log  r» 

log  Wr. 


10  —  n 
10 


(resistarce,  ohms,  copper) 


-— _—  (weight  per  1000  ft., Ibs.  copper; 


158 


TRANSMISSION   LINE    CONSTANTS 


Formulae. 

The  two  following  tables  are  for  one  wire  and  the  neutral 
of  a  three  phase  circuit,  or  for  one  wire  of  a  single  phase  cir- 
cuit. For  both  wires  of  a  single  phase  circuit,  the  reactances 
and  impedances  as  given  must  be  doubled  and  capacities  and 
charging  currents  divided  by  two.  But  for  charging  currents,  it 
must  be  observed  that  for  single  phase  the  e.  m.  f.  is  that  be- 
tween wires  and  for  three  phase  between  one  wire  and  the 
neutral. 

Reactance  per  mile,  one  cycle 

x  =  F  0.00465  log  f  a\  +  0.0019  1    ohms,    ' 

Capacity  per  mile,  C  =  °-038M^  microfarads; 

log  2  a/d- 
in which  a  =  distance  between  wires  and  d  =  diameter  of  wire, 
the  wires  being  arranged  at  the  vertices  of    an    equiangular 
triangle. 

Charging  current  /  =  2  w  n  E  C  10  ~fi  amperes,  where  n  = 
frequency  in  cycles  per  second  and  E  =  e.  in.  f.  between  one 
wire  and  neutral. 

The  table  on  page  160  is  for  two  wires  of  a  single  phase  cir- 
cuit at  25  cycles  per  second. 


Reactance,  Capacity  and  Charging  Current. 

(Per  mile  one  wire  three  phase  at  1  cycle  per  sec.  and  E  -  10000  v.) 


Reactance 
a/d     X 

Capacity 
C 

Charging 
Current 

1 

Reactance 

a/d    x 

Capacity 
C 

Charging 
Current 

1 

25 
30 
35 

.00841 
.00877 
.00908 

.02298 
.02187 
.02112 

.001444 
.001374 
.001327 

150 
175 
200 

.01202 
.01233 
.01260 

.01572 
.01531 
.01497 

.000988 
.000962 
.000940 

40 
45 
50 

.00935 
.00959 
.00978 

.02047 
.01994 
.01947 

.001286 
.001253 
.001223 

250 
300 
350 

.01305 
.01342 
.01373 

.01443 
.01402 
.01369 

.000906 
.000880 
.000860 

60 
70 
80 

.01017 
.01048 
.01075 

.01877 
.0181(5 
.01767 

.001179 
.001141 
.001110 

400 
450 
500 

.01400 
.01424 
.01445 

.01342 
.01318 
.01298 

.000843 
.000828 
.000815 

90 
100 
125 

.01098 
.01120 
.01165 

.01730 
.01693 
.01624 

.001087 
.001064 
.001020 

600 
700 
800 

.01482 
.01513 
.01540 

.01265 
.01238 
.01216 

.000794 
.000777 
.000764 

TRANSMISSION   LINE    CONSTANTS  159 

Capacity,  Charging  Current,  Reactance  and  Impedance. 

(Per  mile  one  wire  three  phase  at  60  cycles  per  sec.  and  E  ~  10000  v.) 


No. 
B.  &S. 

12 

Distance  Between 
18             24             36 

Wires,  Inches. 

48             60 

72 

84 

0000 

C 

.0227 

.0206 

.0193 

.0178 

.0168 

.0161 

.0157 

.0152 

1 

.0855 

.0776 

.0728 

.0670 

.0634 

.0608 

.0591 

.0572 

X 

.509 

.557 

.593 

.642 

.677 

.705 

.726 

.745 

*z 

.575 

.618 

.651 

.696 

.728 

.754 

.774 

.792 

**z 

.668 

.705 

.734 

.774 

.803 

.827 

.845 

.861 

000 

C 

.0220 

.0200 

.0189 

.0174 

.0164 

.0158 

.0153 

.0149 

1 

.0830 

.0756 

.0711 

.0654 

.0620 

.0595 

.0577 

.0562 

X 

.522 

.572 

.607 

.656 

.691 

.718 

.740 

.759 

z 

.622 

.664 

.695 

.738 

.769 

.794 

.814 

.831 

z 

.720 

.790 

.815 

.852 

.879 

.901 

.918 

.934 

00 

C 

.0214 

.0195 

.0184 

.0169 

.0161 

.0155 

.0150 

.0146 

1 

.0807 

.0737 

.0692 

.0640 

.0606 

.0585 

.0566 

.0551 

X 

.538 

.586 

.622 

.671 

.706 

.732 

.754 

.773 

z 

.686 

.724 

.753 

.794 

.824 

.846 

.866 

.882 

z 

.872 

.902 

.926 

.960 

.984 

1.003 

1.020 

1.033 

0 

C 

.0208 

.0191 

.0179 

.0166 

.0158 

.0152 

.0147 

.0144 

1 

.0785 

.0720 

.0676 

.0627 

.0594 

.0572 

.0554 

.0541 

X 

.551 

.601 

.633 

.684 

.719 

.747 

.769 

.787 

z 

.769 

.806 

.830 

.870 

.897 

.920 

.938 

.953 

z 

1.026 

1.053 

1.072 

1.103 

1.125 

1.143 

1.157 

1.170 

1 

C 

.0203 

.0186 

.0175 

.0162 

.0155 

.0149 

.0144 

.0141 

1 

.0765 

.0702 

.0660 

.0612 

.0584 

.0561 

.0544 

.0531 

X 

.565 

.614 

.649 

.699 

.733 

.761 

.783 

.801 

z 

.881 

.913 

.937 

.972 

.997 

1.018 

1.035 

1.045 

z 

1.229 

1.251 

1.269 

1.295 

1.314 

1.330 

1.342 

1.354 

2 

C 

.0198 

.0181 

.0172 

.0159 

.0151 

.0146 

.0142 

.0138 

.0746 

.0685 

.0649 

.0600 

.0572 

.0550 

.0534 

.0522 

X 

.579 

.629 

.663 

.713 

.747 

.774 

.796 

.815 

z 

1.031 

1.060 

1.080 

1.112 

1.134 

1.152 

1.167 

1.180 

z 

1.492 

1.516 

1.530 

1.553 

1.570 

1.584 

1.593 

1.604 

3 

C 

.0193 

.0177 

.0168 

.0157 

.0148 

.0143 

.0139 

.0136 

1 

.0726 

.0668 

.0632 

.0591 

.0560 

.0540 

.0524 

.0512 

X 

.594 

.643 

.678 

.727 

.762 

.788 

.810 

.828 

z 

1.224 

1.248 

1.267 

1.294 

1.314 

1.329 

1.342 

1.355 

z 

1.836 

1.858 

1.863 

1.882 

1.896 

1.907 

1.916 

1.923 

4 

C 

.0188 

.0173 

.0164 

.0153 

.0146 

.0140 

.0137 

.0134 

1 

.0710 

.0654 

.0619 

.0576 

.0549 

.0530 

.0515 

.0504 

X 

.608 

.656 

.692 

.741 

.776 

.803 

.825 

.843 

z 

1.481 

1.501 

1.517 

1.540 

1.557 

1.571 

1.582 

1.593 

z 

2.270 

2.284 

2.295 

2.310 

2.331 

2.338 

2.345 

5 

C 

.0184 

.0169 

.0161 

.0150 

.0143 

.0138 

.0134 

.0131 

1 

.0692 

.0640 

.0606 

.0567 

.0539 

.0521 

.0506 

.0495 

X 

.623 

.671 

.706 

.755 

.789 

.816 

.839 

.857 

z 

1.866 

1.883 

1.896 

1.915 

1.930 

1.940 

1.950 

1.957 

z 

2.828 

2.840 

2.849 

2.861 

2.871 

2.878 

2.885 

2.890 

6 

C 

.0180 

.0166 

.0157 

.0147 

.0140 

.0136 

.0132 

.0129 

1 

.0676 

.0626 

.0594 

.0554 

.0530 

.0512 

.0497 

.0487 

X 

.633 

.685 

.719 

.768 

.803 

.830 

.853 

.871 

z 

2.251 

2.265 

2.274 

2.293 

2.305 

2.313 

2.323 

2.330 

z 

3.541 

3.547 

3.553 

3.564 

3.571 

3.578 

3.583 

3.587 

*  For  copper.     **  For  aluminum.     Other  quantities  are  the  same 
for  either  copper  or  aluminum. 


160  TRANSMISSION  LINE  CONSTANTS 

Capacity,  Charging  Current,  Reactance  and  Impedance. 

(Per  mile  two  wires  single  phase  at  25  cycles  per  sec.  and  E  =  10000  v.) 


No. 
B.  &S. 

12 

Distance  Between  Wires,  Inches- 
18            24             36            48             60 

72 

84 

0000 

C 

.0113 

.0103 

.0097 

.0089 

.0084 

.0081 

.0078 

.0076 

I 

.0178 

.0162 

.0152 

.0140 

.0132 

.0127 

.0123 

.0119 

X 

.424 

.464 

.494 

.535 

.564 

.588 

.605 

.621 

*Z 

.684 

.709 

.729 

.757 

.778 

.796 

.809 

.821 

**Z 

.962 

.981 

.994 

1.015 

1.030 

1.045 

1.055 

1.064 

000 

C 

.0110 

.0100 

.0094 

.0087 

.0082 

.0079 

.0077 

.0075 

I 

.0173 

.0157 

.0148 

.0136 

.0129 

.0124 

.0120 

.0117 

X 

.435 

.477 

.506 

.546 

.576 

.598 

.616 

.632 

z 

.804 

.827 

.844 

.870 

.888 

.903 

.916 

.926 

z 

1.172 

1.189 

1.200 

1.218 

1.231 

1.243 

1.251 

1.259 

00 

C 

.0107 

.0098 

.0092 

.0085 

.0080 

.0078 

.0075 

.0073 

I 

.0167 

.0154 

.0144 

.0133 

.0126 

.0122 

.0118 

.0115 

X 

.448 

.487 

.518 

.559 

.588 

.610 

.628 

.644 

z 

.962 

.980 

.996 

1.017 

1.033 

1.046 

1.056 

1.066 

z 

1.444 

1.456 

1.467 

1.482 

1.494 

1.502 

1.509 

1.516 

0 

C 

.0104 

.0096 

.0090 

.0083 

.0079 

.0076 

.0074 

.0072 

.0161 

.0150 

.0141 

.0131 

.0124 

.0119 

.0115 

.0113 

X 

.458 

.500 

.527 

.569 

.599 

.622 

.640 

.655 

z 

1.164 

1.181 

1.194 

1.211 

1.226 

1.238 

1.247 

1.266 

z 

1.790 

1.800 

1.809 

1.823 

1.831 

1.838 

1.845 

1.851 

1 

C 

.0101 

.0093 

.0088 

.0081 

.0077 

.0074 

.0072 

.0070 

I 

.0160 

.0146 

.0138 

.0128 

.0122 

.0117 

.0113 

.0111 

X 

.471 

.512 

.541 

.582 

.611 

.634 

.652 

.668 

z 

1.430 

1.445 

1.454 

1.470 

1.482 

1.493 

1.500 

1.507 

z 

2.228 

2.238 

2.246 

2.256 

2.264 

2.270 

2.276 

2.280 

2 

C 

.0099 

.0092 

.0086 

.0080 

.0076 

.0073 

.0071 

.0069 

I 

.0156 

.0143 

.0135 

.0125 

.0119 

.0115 

.0111 

.0109 

X 

.483 

.524 

.552 

.594 

.622 

.645 

.664 

.679 

z 

1.758 

1.780 

1.788 

1.802 

1.810 

1.818 

3.825 

1.830 

z 

is.790 

2.800 

2.806 

2.814 

2.820 

2.826 

2.830 

2.832 

3 

C 

.0097 

.0089 

.0084 

.0078 

.0074 

.0072 

.0070 

.0068 

I 

.0151 

.0139 

.0132 

.0123 

.0117 

.0113 

.0109 

.0107 

X 

.495 

.536 

.565 

.605 

.634 

.656 

.675 

.690 

z 

2.197 

2.205 

2.213 

2.2^6 

2.234 

2.240 

2.246 

2.251 

z 

3.501 

3.511, 

3.518 

3.524 

2.529 

3.534 

3.538 

3.535 

4 

C 

.0094 

.0087 

.0082 

.0076 

.0073 

.0070 

.0068 

.0067 

I 

.0148 

.0136 

.0129 

.0120 

.0114 

.0110 

.0107 

.0105 

X 

.506 

.546' 

.576 

.618 

.647 

.668 

.687 

.703 

z 

2.745 

2.753 

2.760 

2.770 

2.776 

2.781 

2.786 

2.790 

z 

4.400 

4.405 

4.410 

t4.417 

4.420 

4.422 

4.424 

4.426 

5 

C 

.0092 

.0085 

.0080 

.0075 

.0072 

.0069 

.0067 

.0066 

I 

.0144 

.0133 

.0126 

.0118 

.0112 

.0108 

.0105 

.0103 

X 

.518 

.559 

.588 

.628 

.657 

.680 

.697 

.714 

z 

3.559 

3.564 

3.569 

3.574 

3.579 

3.583 

3.587 

3.591 

z 

5.544 

5.548 

5.552 

5.555 

5.558 

5.561 

5.564 

5.567 

6 

C 

.0090 

.0083 

.0078 

.0074 

.0070 

.0068 

.0066 

.0065 

I 

.0141 

.0130 

.0124 

,0115 

.0110 

.0107 

.0104 

.0102 

X 

.527 

.571 

.599 

.641 

.669 

.692 

.710 

.725 

z 

4.353 

4.357 

4.361 

4.365 

4.369 

4.373 

4.377 

4.381 

z 

6.980 

6.982 

6.984 

6.987 

6.990 

6.993 

6.996 

6.998 

For   copper.        **  For   aluminum. 


HYDRAULICS 


161 


HYDRAULICS 

Immersed  Rectangle. 
Surface  Level 


Fig.  1 

The  center   of  pressure  for  an  immersed  rectangle,  e.  g.,  a 
dam,  is  given  by 

9     fl  *          rl  ?>  91*    _  7  3 

1        4  a,  —  a,  7_  - 

"  ' 


The  pressure  P  =  7 


J 


pounds     where    7    is     the 


weight  of  a  cubic  foot  of  water  and  A  is  the  area  of  the 
rectangle  in  square  feet. 

Impact  due  to  Jet. 

The  impact  due  to  a  jet  is 

P   =    JL   Q(f—v)    (1  —  cosa) 

9 

in  which  Q  is  the  quantity  per  second  striking  the  bucket  or 
vane,  7  is  the  weight  per  unit  volume  of  the  fluid,  c  is  the 
velocity  of  the  jet,  y.  the  velocity  of  the  vane  in  the  direction  of 
c  and  a  is  the  angle  through  which  the  jet  is  deflected  by  the 


162  HYDRAULICS 

Orifices. 

The  coefficients  of  flow  for  an  orifice  are 
a  '=  coefficient  of  contraction. 
<£  =  coefficient  of  velocity. 
c  =  coefficient  of  efflux, 
f  =  coefficient  of  resistance. 


The  average  values  of  these  coefficients  for  an  orifice  in  a 
thin  plate  are: — 

o=  .62,     0=.98,     <r=.61,     f  =  .041. 

Tables  land  II  give  values  of  c  for  vertical  orifices,  for  var- 
ious heads  and  diameters. 

Partial  contraction  occurs  when  a  portion  of  the  stream  lines 
are  perpendicular  to  the  plane  of  the  orifice  for  some  distance 
inside  the  vessel;  as  when  a  vertical  orifice  is  made  at  the  bot- 
tom of  a  vessel. 

Let  n  =  the  ratio  of  contour  of  inwardly  projecting  guide  to 
the  perimeter  of  the  orifice. 

cn  =  coefficient  of  efflux  for  partial  contraction; 
c0  =  coefficient  of  efflux  for  full  contraction; 
then  <"n  =  ^b  (1  +  kn). 

For  small  circular  orifices,  k=  .128; 

For  square  orifices ,  k  =  .152; 

For  rectangular  orifices.  . ,  k  =  .155. 

Imperfect  contraction  exists  when  water  approaches  the 
orifice  with  considerable  velocity. 


1 

\ 

T 

b 

Fig.  2 

Let  D=  diameter  of  pipe;  d  =  diameter  of  orifice;  d/D=  n; 
cn  =  coefficient  of  efflux;  c0  =  coefficient  of  efflux  for  a  negligible 
velocity  of  approach.  For  circular  openings, 

Cn  =  c0  (1  +.0456  [14.82n  — 1]); 

cn  —  ^    =.0456  [14.82n  — 1]. 

c0 
For  rectangular  openings,  cn  =  c0  (1  +  .076  [9n  —  1]). 

Values  of    tn~Cn--  are  given  in  Table  III. 


HYDRAULICS 
Orifices — Continued 


163 


Fig.  3  Fig.  4 

The  values  of  c  for  orifices  other  than  thin  plates  as  Fig.  3 
and  Fig.  4  are  given  in  Table  IV. 


Fig.  5. 

Table   V  gives  the  values   of  c  for  converging  conical  tubes 
for  a  head  of  about  10  feet  and  about  %"  diameter  at  the  small 


end.       Her  e(Fig.5), 


3d 


Flow  Over  Weirs 
Francis'  formula  for  the  discharge  over  a  rectangular  weir  is, 

Q=cbh~'r 


where  6  is  the  length  and  h  the  head  over  the  crest,  both  ex- 
pressed in  feet.  If  there  are  end  contractions,  b  must  be  di- 
minished by  Vio  h  for  each  such  contraction.  Hence  for  the 
weirs  with  full  contractions  on  both  ends 

Q=c7,,~t^    (&— .2ft). 

c  =  3.33  for  heads  above  0.5  ft.  For  heads  below  0.5  ft.  the 
value  of  c  must  be  taken  from  Table  VI. 

Smith's  formula  for  the  discharge  over  a  weir  is, 


Values  of  c  in  this  formula  are  given  in  Table  VII  for  weirs 
without  end  contractions,  and  in  Table  VIII,  where  there  are 
full  contractions  at  both  ends. 


164  HYDRAULICS 

» 

Flow  Over  Weirs—  Continued 

By  putting  C  =  2/:i  c  -/  2  g,  Smith's  formula  may  be  written 

Q  =  C  b  h  ~* 

Values  of  C  are  given  in  Tables  IX  and  X. 
If  there  is  an  appreciable  velocity  of  approach,    then  for 
Francis'  formula  for  weirs  without  end  contractions, 

Q=  3.33  b  ((h  +  h,)  Jj      —  >i    »  J 
and  for  weirs  with  full  end  contractions, 

Q  =  3.33  (6  —  0.2/1.)   [(h  +  h,  )  ~f~  _  jh    2    ] 

where     /tj  =  -        =  0.01555  #2,  v  being  the  velocity  of  approach. 

29 
For  Smith's  formula, 


where  /jj  is  the  same  as  above,  and  n  =  1.4   for  contracted  weirs 
and  n  —  1  K  for  weirs  without  contractions. 

Table  XI  gives  values  of  h    2      For  rough  estimates  of  weirs 
'Table  XII  may  be  used. 

Flow  Through  Pipes 
The  general  formula  for  the  loss  of  head  in  pipes  is 


in  which  h  is  the  loss  of  head  in  feet,  v  is  the  velocity  in  feet  per 
second  and  £is  a  coefficient. 

The  loss  of  head  by  friction  is 

'    7,  -X    lv* 

^0\ 

where  I  is  the  length  of  the  pipe  in  feet,  d  is  the  diameter  in 
feet  and  X  is  a  coefficient  depending  upon  d  and  v. 

Tables  XIII  and  XV  give  values  of  X  for  cast  iron  and  wood 
stave  pipes  and  Table  XIV  gives  the  loss  of  head  per  1000  feet 
of  pipe  for  cast  iron  when  d  and  v  are  known,  or  when  d  and  the 
discharge  in  cubic  feet  per  minute  are  given. 


HYDRAULICS 

Flow  Through  Pipes— Continued 


165 


Fig.  5  Fig.  6 

For  elbows  f  =  .9457  sin2  V2  5  +  2.047  sin4  y%  8. 

/3° 

For  curvesf=  $"'  - 

90 

f  for  pipes  of  circular  section;   f"  for  rectangular  section. 
Tables  XVI  and  XVII  give  values  of  the  above  coefficients. 

Flow  of  Water  in  Open  Channels 

For  the  flow  of  water  in  ditches,  flumes,   etc.,  the  general 
formula  is 

v  —  c  -j/  rs, 
where  v  is   the  mean  velocity  in  feet  per  sec.,  r  the  hydraulic 


mean  radius  or  depth  =     a  = 


areaof  cross  section 


and  g  .g 


P  wetted  perimeter 

sine  of  the  angle  of  slope  =  h/l.      The  coefficient  c  is  calculated 
by  the  aid  of  Kutter's  formula; 


41.6    + 


1.811  +   .00281 


1  +  (41.6  +  122281) 

8 


V  r 


In  this  formula,  r  and  s  have  the  same  meaning  as  above, 
and  n  is  a  constant  depending  on  the  roughness  of  the  surface. 
If  sides  and  bottom  of  channel  are  lined  with  — 

Well  planed  timber  ...................................  n=  .009 

Neat  cement  (also  for  very  smooth  iron  pipes)    ........  n  =  .010 

Plaster  of  1  sand  and  1  cement  (or  smooth  iron  pipes).  n  =  .011 
Rough  lumber  (ordinary  iron  pipes)  ...............  .  .  .n  =  .012 

Ashlar  or  brickwork  ........................  ..........  n  =  .013 

Rubble  ...............................................  n  =  .017 

Canals  in  very  firm  gravel  ....................  .........  n  =  .020 

Canals  and  weirs  in  moderately  good  order  .............  n  =  .025 

Canals  in  bad  order  ...................................  n  =  .035 


Table  XVIII  gives   values  of  rand  c    \/  r  for  various  values 


of 


166 


HYDRAULICS 


Table   I. — Coefficients  of  Efflux  for  Circular  Vertical   Orifices. 

HAMILTON  SMITH,   JR. 


Head  in 
Feet 

0.02 

0.04 

Diameter  of  Orifice  in  Feet 
0.07             0.1              0.2 

0.6 

1 

0.4 

.637 

.624 

.618 

0.6 

.655 

.630 

.618 

.613 

.601 

.593 

0.8 

.648 

.626 

.615 

.610 

.601 

.594 

.590 

1.0 

.644 

.623 

.612 

.608 

.600 

.595 

.591 

1.5 

.637 

.618 

.608 

.605 

.600 

.596 

.593 

2 

.632 

.614 

.607 

.604 

.599 

.597 

.595 

3 

.627 

.611 

.604 

.603 

.599 

.598 

.597 

4 

.623 

.609 

.603 

.602 

.599 

.597 

.596 

6 

.618 

.607 

.602 

.600 

.598 

.597 

.596 

8 

.614 

.605 

.601 

.600 

.598 

.596 

.596 

10 

.611 

.603 

.599 

.598 

.597 

.596 

.595 

20 

.601 

.599 

.597 

.59& 

.596 

.596 

.594 

50(?) 

.596 

.595 

.594 

.594 

.594 

.594 

.593 

100  (?) 

.593 

.592 

.592 

.592 

.592 

.592 

.592 

Table 

11.  —  Coefficients 

of   Efflux 

for  Square  Vertical 

Orifices. 

HAMILTON  SMITH.   JR. 

Head  in 
Feet 

Side  of  Square  in 

Feet 

0.02 

0.04 

0.07 

0.1 

0.2 

0.6 

1 

0.4 

.643 

.628 

.621 

0.6 

.660 

.636 

.623 

.617 

.605 

.598 

0.8 

.652 

.631 

.620 

.615 

.605 

.600 

.597 

1.0 

.648 

.628 

.618 

.613 

.605 

.601 

.599 

1   F 

.641 

.622 

.614 

.610 

.605 

.602 

.601 

2 

.637 

.619 

.612 

.608 

.605 

.604 

.602 

3 

.632 

.616 

.609 

.607 

.605 

.604 

.603 

4 

.628 

.614 

.608 

.606 

.605 

.603 

.602 

6 

.623 

.612 

.607 

.605 

.604 

.603 

.602 

8 

.619 

.610 

.606 

.605 

.604 

.603 

.602 

10 

.616 

.608 

.605 

.604 

.603 

.602 

.601 

20 

.606 

.604 

.602 

.602 

.602 

.601 

.600 

50(?) 

.602 

.601 

.601 

.600 

.600 

.599 

.599 

100(?) 

.599 

.598 

.598 

.598 

.598 

.598 

.598 

HYDRAULICS  167 

Table  III — Imperfect  Contraction 


Cn—Co 

Cn-C 

'o 

Co 

Co 

•n                     C 

R 

n                     c 

R 

.05                    .007 

.009 

.55                      .161 

.178 

.10               .or4 

.019 

.60                      .189 

.208 

.15                     .023 

.030 

.65                      .223 

.241 

.20                    .034 

.042 

.70                      .260 

.278 

.25                    .045 

.056 

.75                      .303 

.319 

.30                    .059 

.071 

.80                      .351 

.365 

.35                     .075 

.088 

.85                      .408 

.416 

.40                     .092 

.107 

.90                      .471 

.473 

.45                     .112 

.128 

.95                      .546 

.537 

.50                     .134 

.152 

1.00                     .631 

.608 

C  —  Circular 

,  R  —  Rectangular  Openings. 

Table   IV—  Coefficient   (c) 

for  Conical   Orifices 

5         0°         11°        22° 

45°         67° 

90°      112°      135°      157° 

180° 

C          .96         .92         .90 

.75         .68 

.63         .60         .58         .54 

.54 

Table  V—  Coefficients 

for  Conical  Tubes 

Angle  of  Cone 

C 

0                       a 

0°                00' 

.829 

.829                  1.000 

1                  36 

.866 

.867                 1.000 

4            .     10 

.912 

.910                 1.000 

7                 52 

.930 

.932                   .998 

10                 20 

.938 

.951                   .986 

13                 24 

.946 

.963                   .983 

16                 36 

.938 

.971                   .966 

21                 00 

.919 

.972                   .945 

29                  58 

.895 

.975                   .918 

48                 50 

.847 

.984                   .861     • 

168  HYDRAULICS 

Table  VI — Discharge  Coefficients  for  Rectangular  Weirs 

FRANCIS'  FORMULA. 


EffecHve  head  // 

Coefficient  c 

Effective  head  h 

Coefficient  c 

0.5  and  over 
0.4 
0.3 
0.25 

3.33 
3.337 
3.353 

3.368 

0.2 
0.15 
0.1 
0.06 

3.338 
3.430 
3.528 
3.750 

Table    VII — Discharge     Coefficients    for     Rectangular     Weirs 

SMITH'S  COEFFICIENTS  FOR  WEIRS  WITHOUT 
END  CONTRACTIONS 


Effective 
head  h       2 

Length  of  Weir  in  Feet. 
3457 

10 

19 

0.1 

0.659 

0.658 

0.658 

0.657 

0.15 

0.652 

0.649 

0.647 

.645 

.645 

.644 

.643 

0.2 

.645 

.642 

.641 

.638 

.637 

.637 

.635 

0.25 

.641 

.638 

.636 

.634 

.633 

.632 

.630 

0.3 

.639 

.636 

.633 

.631 

.629 

.628 

.626 

0.4 

.636 

.633 

.630 

.628 

.625 

.623 

.621 

0.5 

.637 

.633 

.630 

.627 

.624 

.621 

.619 

0.6 

.638 

.634 

.630 

.627 

.623 

.620 

.618 

0.7 

.640 

.635 

.631 

.628 

.624 

.620 

.618 

0.8 

.643 

.637 

.633 

.629 

.625 

.621 

.618 

0.9 

.645 

.639 

.635 

.631 

.627 

.622 

.619 

1.0 

.648 

.641 

.637 

.633 

.628 

.624 

.619 

1.2 

.646 

.641 

.636 

.632 

.626 

.620 

1.4 

.644 

.640 

.634 

.629 

.622 

1.6 

.647 

.642 

.637 

.631 

.623 

Table    VIM — Discharge    Coefficients    for    Rectangular    Weirs 

SMITH'S  COEFFICIENTS  FOR  WEIRS  WITH 
END  CONTRACTIONS 


Effective 

Length  of  Weir  in  Feet. 

head  h 

0.66 

1 

2 

3 

5 

10 

19 

0.1 

0.632 

0.639 

0.646 

0.652 

0.653 

0.655 

0.656 

0.15 

.619 

.625 

.634 

.638 

.640 

.641 

.642 

0.2 

.611 

.618 

.626 

.630 

.631 

.633 

.634 

0.25 

.605 

.612 

.621 

.624 

.626 

.628 

.629 

0.3 

.601 

.608 

.616 

.619 

.621 

.624 

.625 

0.4 

.595 

.601 

.609 

.613 

.615 

.618 

.620 

0.5 

.590 

.596 

.605 

.608 

.611 

.615 

.617 

0.6 

.587 

.593 

.601 

.605 

.608 

.613 

.615 

0.7 

.590 

.598 

.603 

.606 

.612 

.614 

0.8 

.595 

.600 

.604 

.611 

.613 

0.9 

.592 

.598 

-603 

.609 

.612 

1.0 

.590 

.595 

.601 

.608 

.611 

1.2 

.585 

.591 

.597 

.605 

.610 

1.4 

.580 

.587 

.594 

.602 

.609. 

1.6 

.582 

.591 

.600 

.607 

HYDRAULICS  169 

Table  IX.     Discharge  Coefficient  C  for  Rectangular  Weirs. 

From  Smith's  Coefficients  for  Weirs  without  end  contractions. 


Effective 

Length  of  Weir  in  Feet. 

head  h 

2 

3 

4                  5 

7 

10 

19 

0.1 

3.526 

3.520 

3.520 

3.515 

0.15 

3.488 

3.472 

3.461           3.451 

3.451 

3.445 

3.440 

0.2 

3.450 

3.435 

3.429           3.413 

3.408 

3.408 

3.397 

0.25 

3.429 

3.413 

3.403           3.392 

3.386 

3.381 

3.371 

0.3 

3.418 

3.403 

3.386           3.376 

3.365 

3.360 

3.349 

0.4 

3.403 

3.386 

3.371           3.360 

3.344 

3.333 

3.322 

0.5 

3.408 

3.386 

3.371           3.354 

3.338 

3.322 

3.312 

0.6 

3.413 

3.392 

3.371           3.354 

3.333 

3.317 

3.306 

0.7 

3.424 

3.397 

3.376           3.360 

3.338 

3.317 

3.306 

0.8 

3.441 

3.408 

3.386           3.365 

3.344 

3.322 

3.306 

0.9 

3.451 

3.418 

3.397           3.375 

3.354 

3.328 

3.312 

1.0 

3.467 

3.429 

3.408           3.386 

3.360 

3.338 

3.312 

1.2 

3.456 

3.429           3.403 

3.381 

3.349 

3.317 

1.4 

3.445           3.424 

3.392 

3.365 

3.328 

1.6 

3.461           3.435 

3.408 

3.376 

3.333 

Table  X. 

Discharge  Coefficient  C  for 

Rectangular  Weirs. 

From  Smith's  Coefficients  for  Weirs  with  end  contractions. 

Effectiv< 

l                                            Length  of  Weir  in  Feet 

head  //, 

0.66 

1 

2                  3 

5 

10 

19 

0.1 

3.381 

3.419 

3.456           3.488 

3.494 

3.504 

3.510 

0.15 

3.312 

3.344 

3.392           3.413 

3.424 

3.429 

3.435 

0.2 

3.269 

3.306 

3.349           3.371 

3.376 

3.386 

3.392 

0.25 

3.237 

3.274 

3.322           3.338 

3.349 

3.360 

3.365 

0.3 

3.215 

3.253 

3.296           3.312 

3.322 

3.338 

3.344 

0.4 

3.183 

3.215 

3.258           3.280 

3.290 

3.306 

3.317 

0.5 

3.156 

3.189 

3.237           3.253 

3.269 

3.290 

3.301 

0.6 

3.140 

3.172 

3.215           3.237 

3.253 

3.280 

3.290 

0.7 

3.130 

3.156 

3.199           3.226 

3.242 

3.274 

3.285 

0.8 

3.183           3.215 

3.231 

3.269 

3.280 

0.9 

3.167           3M99 

3.226 

3.258 

3.274 

1.0 

3.156           3.183 

3.215 

3.253 

3.269 

1.2 

3.130           3.162 

3.194 

3.237 

3.264 

1.4 

3.103           3.140 

3.178 

3.221 

3.258 

1.6 

3.114 

3.162 

3.210 

3.247 

:i 

Table  XI. 

Values  of  h    2    for 

h  =  0 

to  2.95 

h 

0 

05 

10 

15             20           25 

30 

35 

40 

45 

0 

0 

.0112 

.0316 

.0581       .0894       .1250 

.1643 

.2071 

.2530 

.3019 

1 

1.0000 

1.0759 

1.1537 

1.2332     1.3145     1.3975 

1.4822 

1.56.S6 

1.6565     1 

,7460 

2 

2.8284 

2.9352 

3.0432 

3.1525     3.2631     3.3750 

3.4881 

3.6025 

3.7181     3.8349 

50 

55 

60 

65             70           75 

80 

85 

90 

95 

0 

.3536 

.4079 

.4648 

.5240       .5857       .6495 

.7155 

.7837 

.8538 

.9259 

1 

1.8371 

1.9297 

2.0238 

2.1195     2.2165     2.3150 

2.4150 

2.5163 

2.6190     2.7230 

2 

3.9529 

4.0720 

4.1924 

4.3139     4.4366     4.5604 

4.6853 

4.8114 

4.9385     5.0668 

170  HYDRAULICS 


Table  XII — Weir  Measurement 

PELTON  WATER  WHEEL  CO. 

Giving  Cubic  Feet  of  Water  per  minute  that  will  flow  over  a 
Weir  1  inch  wide  and  from  %  to  20%  inches  deep. 


Inches                '/s               !/4               %               !/2              %              % 

7/8 

.0 

.00 

.01 

.05 

.09 

.14 

.19 

.26 

.32 

1 

.40 

.47 

.55 

.64 

.73 

.82 

.92 

1.02 

2 

1.13 

1.23 

1.35 

1.46 

1.58 

1.70 

'1.82 

1.95 

3 

2.07 

2.21 

2.34 

2.48 

2.61 

2.76 

2.90 

3.05 

4 

3.20 

3.35 

3.50 

3.66 

3.81 

3.97 

4.14 

4.30 

5 

4.47 

4.64 

4.81 

4.98 

5.15 

5.33 

5.51 

5.69 

6 

5.87 

6.06 

6.25  , 

6.44 

6.62 

6.82 

7.01 

7.21 

7 

7.40 

7.60 

7.80 

8.01 

8.21 

8.42 

8.63 

8.83 

8 

9.05 

9.26 

9.47 

9.69 

9.91 

10.13 

10.35 

10.57 

9 

10.80 

11.02 

11.25 

11.48 

11.71 

11.94 

12.17 

12.41 

10 

12.64 

12.88 

13.12 

13.36 

13.60 

13.85 

14.09 

14.34 

11 

14.59 

14.84 

15.09 

15.34 

15.59 

15.85 

16.11 

16.36 

12 

16.62 

16.88 

17.15 

17.41 

17.67 

17.94 

18.21 

18.47 

13 

18.74 

19.01 

19.29 

19.56 

19.84 

20.11 

20.39 

20.67 

14 

20.95 

21.23 

21.51 

21.80 

22.08 

22.37 

22.65 

22.94 

15 

23.23 

23.52 

23.82 

24.11 

24.40 

24.70 

25.00 

25.30 

16 

25.60 

25.90 

26.20 

26.50 

26.80 

27.11 

27.42 

27.72 

17 

28.03 

28.34 

28.65 

28.97 

29.28 

29.59 

29.91 

30.22 

18 

30.54 

30.86 

31.18 

31.50 

31.82 

32.15 

32.47 

32.80 

19 

33.12 

33.45 

33.78 

34.11 

34.44 

34.77 

35.10 

35.44 

20 

35.77 

36.11 

36.45 

36.78 

37.12 

37.46 

37.80 

38.15 

HYDRAULICS 


171 


Table  XIII— Coefficients  of  Flow   (X)   of  Water  in  Clean   Iron 


Pipes  Under  Pressure 


Velocity 

Diameters 

ft.  per      !/2" 

1" 

2" 

3" 

4" 

6" 

8" 

10" 

12" 

14" 

sec. 

.0417' 

.0833' 

.1667' 

.2500' 

.3333' 

.5000' 

.  6667' 

.8333' 

1.000' 

1.167' 

.1 

.0480 

.0456 

.0432 

.0422 

.0404 

.0370 

.0344 

.0320 

.0304 

.0292 

.3 

.0443 

.0428 

.0396 

.0380 

.0360 

.0336 

.0312 

.0296 

.0279 

.0268 

.5 

.0418 

.0398 

.0364 

.0354 

.0340 

.0317 

.0296 

.0283 

.0268 

.0256 

.7 

.0402 

.0372 

.0348 

.0335 

.0322 

.0303 

.0285 

.0272 

.0260 

.0248 

1. 

.0381 

.0353 

.0330 

.0317 

.0306 

.0289 

.0274 

.0262 

.0250 

.0241 

1.5 

.0356 

.0332 

.0312 

.0299 

.0289 

.0274 

.0261 

.0249 

.0239 

.0231 

2. 

.0340 

.0317 

.0301 

.0288 

.0279 

.0264 

.0253 

.0242 

.0233 

.0225 

2.5 

.0327 

.0308 

.0292 

.0280 

.0271 

.0257 

.0246 

.0236 

.0228 

.0220 

3. 

.0317 

.0300 

.0284 

.0273 

.0265 

.0252 

.0242 

.0232 

.0224 

.0217 

3.5 

.0308 

.0292 

.0277 

.0267 

.0259 

.0247 

.0237 

.0228 

.0221 

.0214 

4. 

.0300 

.0285 

.0272 

.0262 

.0255 

.0243 

.0233 

.0225 

.0217 

.0211 

5. 

.0287 

.0274 

.0263 

.0254 

.0247 

.0236 

.0227 

.0220 

.0213 

.0207 

6. 

.0276 

.0266 

.0255 

.0248 

.0241 

.0231 

.0223 

.0216 

.0210 

.0203 

7. 

.0268 

.025& 

.0249 

.0243 

.0236 

.0227 

.0219 

.0212 

.0207 

.0201 

8. 

.0261 

.0253 

.0244 

.0238 

.0232 

.0224 

.0216 

.0210 

.0205 

.0199 

10. 

.0250 

.0245 

.0237 

.0232 

.0226 

.0219 

.0212 

.0206 

.0201 

.0196 

12. 

.0244 

.0239 

.0232 

.0228 

.0223 

.0216 

.0210 

.0204 

.0199 

.0194 

16. 

.0235 

.0232 

.0226 

.0220 

.0218 

.0211 

.0205 

.0200 

.0194 

.0191 

20. 

.0231 

.0228 

.0223 

.0219 

.0214 

.0208 

.0201 

.0197 

.0192 

.0188 

Velocity 

Diameters 

ft.  per     16" 

18" 

20" 

24" 

30" 

36" 

44" 

48" 

54" 

60" 

sec. 

1.333' 

1  .  500' 

1  .  667' 

2.000' 

2.500' 

3.000' 

3.667 

4.000' 

4.500' 

5.000' 

0.1 

.0280 

.0268 

.0260 

.0246 

.3 

.0256 

.0248 

.0240 

.0225 

.5 

.0244 

.0236 

.0229 

.0212 

.0194 

.0177 

.0160 

.0153 

.0144 

.0136 

.7 

.0238 

.0229 

.0221 

.0207 

.0190 

.0174 

.0158 

.0152 

.0143 

.0136 

1. 

.0232 

.0224 

.0216 

.0202 

.0186 

.0171 

.0156 

.015*0 

.0142 

.0135 

1.5 

.0224 

.0216 

.0209 

.0196 

.0182 

.0168 

.0154 

.0148 

.0140 

.0134 

2. 

.0218 

.0211 

.0204 

.0193 

.0179 

.0166 

.0153 

.0147 

.0140 

.0134 

2.5 

.0213 

.0207 

.0201 

.0190 

.0177 

.0165 

.0152 

.0146 

.0139 

.0132 

3. 

.0210 

.0204 

.0198 

.0187 

.0175 

.0164 

.0151 

.0145 

.0138 

.0132 

3.5 

.0207 

.0201 

.0196 

.0186 

.0174 

.0163 

.0150 

.0145 

.0138 

.0132 

4. 
5. 

.0206 
.0201 

!om 

.0194 
.0191 

.0184 
.0182 

.0173 
.0171 

.0162 
.0161 

.0150 
.0149 

.0145 
.0144 

.0137 
.0137 

.0131 
.0131 

6 

.0198 

.0193 

.0189 

.0180 

.0169 

.0160 

.0148 

.0143 

.0136 

.0130 

7. 

.0196 

.0191 

.0187 

.0178 

.0168 

.0158 

.0147 

.0142 

.0135 

.0130 

8. 

.0194 

.0190 

.0186 

.0177 

.0167 

.0157 

.0147 

.0142 

.0135 

.0129 

10. 

.0192 

.0188 

.0184 

.0176 

.0166 

.0156 

.0146 

.0141 

.0134 

.0128 

12. 

.0190 

.0186 

.0182 

.0174 

.0164 

.0155 

.0145 

.0140 

.0133 

.0128 

16. 

.0187 

.0182 

.0180 

.0172 

.0162 

.0154 

.0143 

.0139 

.0132 

.0126 

20. 

.0184 

.0181 

.0177 

.0170 

.0161 

.0152 

.0142 

.0138 

.0130 

.0126 

172  HYDRAULICS 

TabJe    XIV — Loss   of    Head    by    Friction    in     Clean    Iron    Pipes 


Dia- 

Area 

Velocity 

in  Feet  per  Second. 

meter    sq. 

1 

' 

M/2 

2 

3 

in. 

ft. 

Q 

F 

Q 

F 

Q 

F 

Q 

F 

1 

.  0055 

.33 

7.00 

.50 

.   12.9 

.66 

24.2 

.99 

49.3 

2 

.0218 

1.31 

3.16 

1.96 

6.55 

2  62 

11.3 

3.92 

23.6 

3 

.0491 

2  95 

2.00 

4.42 

4.20 

5!  89 

7.15 

8.84 

15.1 

4 

.0893 

5.23 

1.47 

7.84 

3.04 

10.5 

5.18 

15.7 

11.1 

6 

.1963 

11.8 

.93 

17.7 

1.92 

23.5 

3.27 

35.3 

7.00 

8 

.3491 

20.9 

.63 

31.3 

1.37 

41.8 

2.36 

62.8 

5.08 

10 

.5454 

32.7 

.50 

49.0 

1.05 

65.4 

1.80 

98.2 

3.89 

12 

.7854 

47.1 

.40 

70.6 

.83 

94.2 

1.44 

141 

3.14 

14 

1.069 

64.1 

.33 

96.1 

.69 

128 

1.20 

192 

2.60 

16 

1.396 

83.7 

.29 

126 

.59 

167 

1.13 

251 

2.21 

18 

1.768 

106 

.24 

159 

.50 

212 

.87 

318 

1.90 

20 

2.182 

131 

.21 

195 

.44 

261 

.76 

393 

1.66 

24 

3.142 

188 

.16 

282 

.34 

377 

.60 

565 

1.31 

30 

4.909 

294 

.12 

441 

.25 

589 

.44 

883 

.98 

36 

7.069 

424 

.091 

636 

.20 

848 

.34 

1272 

.77 

44 

10.56 

634 

.068 

951 

.15 

1267 

.  26 

1900 

.58 

48 

12.57 

754 

.060 

1131 

.13 

1508 

.23 

2262 

.51 

54 

15.90 

954 

.050 

1431 

.11 

1909 

.19 

2863 

.43 

60 

19.64 

1178 

.043 

1767 

.094 

2356 

.17 

3534 

.37 

Dia 

-    Area 

Velocity  in  Feet  per  Second 

met* 

;r     sq- 

4 

5 

6 

7 

in. 

ft. 

Q 

F 

Q 

F 

Q 

F 

Q 

F 

1 

.0055 

1.32 

83.8 

1.65 

127 

1.98 

180 

2.31 

240 

2 

.0218 

5.23 

39.0 

6.54 

61.2 

7.85 

85.2 

9.16 

114 

3 

.0491 

11.7 

26.0 

14.7 

39.4 

17.7 

55.1 

20.5 

73.7 

4 

.0893 

20.9 

19.0 

21  2 

28.8 

31.4 

40.4 

36.7 

53.9 

6 

.1963 

47.1 

12.1 

58.9 

18.2 

70.6 

25.9 

82.4 

34.5 

8 

.3491 

83.7 

8.31 

104 

13.2 

126 

18.7 

146 

25.0 

10 

.5454 

131 

6.70 

163 

10.3 

196 

14.5 

229 

19.3 

12 

.7854 

188 

5.37 

235 

8.26 

283 

11.8 

330 

15.7 

14 

1.069 

257 

4.49 

321 

6.89 

385 

9.74 

449 

13.1 

16 

1.396 

335 

3.84 

419 

5.86 

502 

8.34 

586 

11.2 

18 

1.768 

424 

3.27 

530 

5.08 

636 

7.11 

742 

9.65 

20 

2.182 

523 

2.89 

654 

4.42 

785 

6.3^ 

916 

8.51 

24 

3.142 

753 

2.28 

942 

3.54 

1131 

5'V 

-    1320 

6.76 

30 

4.909 

1177 

1.72 

1472 

2.65 

1767 

3.80 

2062 

5.11 

36 

7.069 

1696 

1.34 

2120 

2.10 

2544 

2.97 

2969 

4.03 

44 

10.56 

2534 

1.01 

3168 

1.58 

3801 

2.26 

4435 

3.04 

48 

12.57 

3015 

.89 

3760 

1.40 

4524 

2.00 

5278 

2.70 

54 

15.90 

3816 

.75 

4771 

1.18 

5725 

1.69 

6680 

2.28 

60 

19.64 

4712 

.65 

5890 

1.02 

7068 

1.46 

8246 

1.98 

Q — Discharge  in  cubic  feet  per  minute.     F — Head  loss  in  feet 
per  1000  ft  of  pipe.    Friction  in  riveted  pipe  =  1.2F. 


HYDRAULICS 


173 


Table  XV— Coefficients  of  Flow    (X)    of  Water  in  Wood  Stave 
Pipes  Under  Pressure 


Velocity 
ft-  per 
sec. 

14" 
1.167' 

16" 
1.333' 

Diameters 
18"     20" 
1.500'   1.667' 

22" 
1.833' 

24" 
2' 

30" 
2.5' 

.2 
.5 
.7 

.0379 
.0229 
.0214 

.0415 
.  0232 
.0216 

.0494 
.0243 
.0219 

.0262 
.0229 

.0281 
.0237 

.  0301 
.0247 

.0398 
.0294 

1.0 
1.5 
2.0 

.0206 
.0202 
.0201 

.0206 
.0201 
.0199 

.0207 
.0200 
.0198 

.0210 
.0200 
.0198 

.0213 
.0201 
.0197 

.0218 
.0202 
.0197 

.0239 
0210 
.0199 

2.5 
3.0 
3.5 

.0201 
.0200 
.0200 

.0199 
.0198 
.0198 

.0197 
.0196 
.0196 

.0196 
.0195 
.0195 

.0195 
.0194 
.0193 

.0194 
.0193 
.0192 

.0194 
.0192 
.0190 

4.0 
5.0 
6.0 

.0199 
.0199 
.0199 

.0198 
.0197 
.0197 

.0196 
.0195 
.0195 

.0195 
.0194 
.0194 

.0193 
.0192 
.0192 

.0192 
.0191 
.0191 

.0189 
.0188 
.0187 

7.0 
8.0 
10.0 

.0199 
.0199 
.0199 

.0197 
.0197 
.0197 

.0195 
.0195 
.0195 

.0194 
.0194 
.0194 

.0192 
.0192 
.0192 

.0191 
.0190 
.0190 

.0187 
.0187 
.0187 

12.0 
16.0 
20.0 

.0199 
.0199 
.0199 

.0197 
.0197 
.0197 

.0195 
.0195 
.0195 

.0194 
.0194 
.0194 

.0192 
.0192 
.0192 

.0190 
.0190 
.0190 

.0186 
.0186 
.0186 

Velocity 
ft.  per 
sec. 

36" 
3' 

42" 
3.5' 

Diameters 
48"     54" 
4'      4.5' 

60" 
5' 

66" 
5.5' 

72" 
6' 

.5 
.7 
1.0 

.0499 
.0344 
.0261 

.  .0398 
.0285 

.0360 
.0260 

.0259 
.0203 

.0250 
.0200 

.0239 
.0197 

.0222 
.0190 

1.5 
2.0 
2.5 

.0217 
.0202 
.0195 

.0225 
.0204 
.0194 

.0207 
.0188 
.0180 

.0173 
.0162 
.0158 

.0173 
.0164 
.0160 

.0173 
.0166 
.0163 

.0174 
.0168 
.0166 

3.0 
3.5 

4.0 

.0191 
.0189 
.0187 

.0188 
.0185 
.0183 

.0175 
.0172 
.0170 

.0155 
.0154 
.0152 

.0157 
.0156 
.0155 

.0161  - 
.0160 
.0159 

.0164 
.0163 
.0163 

5.0 
6.0 
7.0 

.0185 
.0184 
.0184 

.0181 
.0179 
.0179 

.0168 
.0167 
.0166 

~?0151 

.0150 
.0150 

.0154 
.0154 
.0153 

.0158 
.0157 
.0157 

.0162 
.0162 
.0162 

8.0 
10.0 
12.0 

.0183 
.0183 
.0183 

.0178 
.0177 
.0177 

.0166 
.0165 
.0165 

.0150 
.0150 
.0149 

.0153 
.0152 
.0152 

.0157 
.0157 
.0157 

.0161 
.0165 
.0161 

16.0 
20.0 

.0183 
.0182 

.0177 
.0177 

.0165 
.0164 

.0149 
.0149 

.0152 
.0152 

.0156 
.0156 

.0161 
.0161 

3  7-i  HYDRAULICS 

Table   XVI — Coefficient  of   Friction  for   Elbows 


5 

r 

20° 
.046 

40° 
.139 

60° 
.364 

80° 

.740 

90° 

.984 

100° 
1.260 

120° 
1.861 

140° 
2.4S1 

Table  XVII — Coefficient  of  Friction  for  Curves 


d/r 

.4 

.6 

.8 

1 

1.2 

1.4 

1.6 

1.8 

2 

r 

.138 

.158 

.206 

.294 

.440 

.661 

.977 

1.408 

1.979 

r 

.14 

.18 

.25 

.40 

.64 

1.02 

1.55 

2.27 

3.23 

Circular  Section. 


"  Rectangular  Section. 


Table  XVIII.     Kutter's  Coefficients, 
n  -  0.009 


1  in  20000 

*J  r  c   c\/r 

1  in 
c 

10000 

ft/r 

1  in 
c 

5000 

c  vxr 

1  in 

c 

2500 

c  v/r 

1  in  1000 

c  cv'r 

A 

93 

37 

106 

42 

114 

46 

119 

48 

123 

49 

.5 

108 

54 

120 

60 

128 

64 

133 

67 

137 

68 

.6 

121 

73 

132 

79 

140 

84 

145 

87 

147 

88 

.7 

133 

93 

143 

100 

150 

105 

154 

108 

156 

109 

.8 

142 

114 

152 

122 

158 

126 

162 

129 

164 

131 

.9 

151 

136 

160 

144 

165 

149 

168 

151 

170 

153 

1.0 

159 

159 

167 

167 

171 

171 

174 

174 

175 

175 

1.2 

173 

207 

178 

214 

181 

218 

183 

220 

184 

221 

1.5 

188 

283 

191 

287 

193 

289 

193 

290 

194 

291 

2.0 

208 

415 

206 

413 

205 

411 

205 

410 

205 

409 

2.5 

221 

553 

217 

541 

214 

535 

213 

532 

212 

529 

3.0 

231 

694 

224 

673 

220 

660 

218 

654 

217 

650 

n  -  0.010 


v/r 

1  in  20000 
c   c\/r 

1  in 

c 

10000 
rv/r 

1  in 

c 

5000 
<rv7  r 

1  in 

c 

2500 
cVr 

1  in  1000 
c  c  V7r 

.4 

81 

32 

91 

37 

99 

40 

104 

42 

107 

43 

.5 

94 

47 

105 

52 

112 

56 

117 

58 

120 

60 

.6 

106 

64 

116 

70 

123 

74 

127 

76 

130 

78 

.7 

116 

81 

126 

88 

132 

92 

136 

95 

138 

97 

.8 

126 

100 

134 

107 

140 

112 

143 

114 

145 

116 

.9 

134 

120 

141 

127 

146 

132 

149 

134 

151 

136 

1.0 

141 

141 

148 

148 

152 

152 

155 

155 

156 

156 

1.2 

154 

184 

159 

190 

162 

194 

164 

196 

165 

198 

1.5 

169 

253 

171 

257 

173 

259 

174 

260 

174 

261 

2.0 

187 

375 

186 

372 

185 

370 

185 

370 

185 

369 

2.5 

201 

501 

196 

490 

194 

484. 

192 

481 

192 

479 

3.0 

210 

631 

204 

611 

200 

599 

198 

593 

196 

589 

HYDRAULICS  175 

Kutter's  Coefficients— Con' t. 
n  =  0.011 


1  in 

20000 

1  in 

10000 

1  in 

5000 

1  in 

2500 

1  in 

1000 

Vr 

C 

c\/  r 

c 

c\/  r 

c 

cVr 

c 

c\/  r 

c  cV'r 

A 
.5 
.6 

71 

83 
94 

28 
42 
56 

80 
92 
103 

32 
46 
62 

87 
99 
109 

35 

49 
65 

91 
103 
113 

36 
52 

68 

94 

106 
115 

38 
53 

t>9 

.7 
.8 
.9 

104 
112 
120 

72 
90 
108 

112 
120 
127 

78 
96 
114 

118 
125 
131 

82 
100 
118 

121 
128 
134 

85 
102 
120 

123 
130 
136 

86 
104 
122 

1.0 
1.2 
1.5 

126 
138 
153 

126 
166 
229 

133 
143 
155 

133 
172' 
233 

137 
146 
156 

137 
175 
235 

139 
148 
157 

139 
177 
236 

140 
149 
158 

140 
178 
237 

2.0 
2.5 
3.0 

171 
184 
193 

341 
459 

'  580 

169 
179 

187 

339 
448 
560 

169 
177 
183 

337 

442 
549 

168 
176 
181 

337 
439 
543 

168 
175 

180 

336 
437 
539 

n 

=  0.012 

1  in 

20000 

1  in 

10000 

1  in 

5000 

1  in 

2500 

1  in 

1000 

Vr 

c 

C  \/  T 

c 

c\/r 

c 

cVr 

c 

cVr 

c  c 

•Vr 

.4 
.5 
.6 

63 

74 

84 

25 
37 
50 

71 
83 
92 

28 
41 
55 

77 
88 
98 

31 

44 
59 

81 
92 
101 

32 

46 
61 

84 
95 
,103 

34 

47 
62 

.7 
.8 
.9 

93 
101 

108 

65 

81 
97 

101 
108 
114 

70 
86 
103 

106 
113 
119 

74 
90 

107 

109 
115 
121 

76 
92 
109 

111 
117 
123 

78 
94 
110 

1.0 
1.2 
1.5 

114 
126 
140 

114 
151 

209 

120 
130 
142 

120 
156 
212 

124 
133 
143 

124 
159 
214 

126 
134 
144 

126 
161 
216 

127 
135 
144 

127 
162 
216 

2.0 
2.5 
3.0 

157 
169 
179 

314 
423 
536 

156 
165 
173 

311 
413 

518 

155 
163 
169 

'310 
408 
507 

155 
162 
167 

309 
405 
501 

154 
161 
166 

308 

403 
498 

n 

=  0  013 

1  in 

20000 

1  in 

10000 

1  in 

5000 

1  in 

2500 

1  in 

1000 

\/r 

c 

C  V  T 

c 

cVr 

c 

c\/  r 

c 

cV  r    c  c\/  r 

A 
.5 
.6 

57 
67 
76 

23 
33 
46 

64 
74 
83 

26 
37 
50 

69 

80 
88 

28 
40 
53 

73 
83 
92 

29 
41 
55 

75 
85 
94 

30 
43 
56 

.8 
1.0 
1.2 

92 
104 
115 

73 
104 
138 

98 
110 
119 

78 
110 
143 

102 
113 
122 

82 
113 
146 

105 
115 
123 

84 
115 

148 

107 
'  117 
124 

85 
117 
149 

1.5 
2.0 
2.5 

128 
145 
157 

193 
290 
393 

130 
144 
153 

196 
288 
384 

132 
143 
151 

197 
286 
378 

132 
.143 
150 

198 
286 
375 

133 
142 
149 

199 
285 
373 

3.0   167    500    161    482    157    471    155    466    154   462 


176 


HYDRAULICS 

Kutter's  Coefficients— Con't. 
n  *  0.017 


1  in 

20000 

1  in 

10000 

1  in 

oOOO 

1  in 

2500 

1  in 

1000 

Vr 

c 

cVr 

c 

cV  r 

c 

cVr 

c 

cVr 

c  c 

V  r 

A 
.5 
.6 

40 
47 
54 

16 
24 
32 

45 
52 
59 

18 
26 
35 

48 
56 
63 

19 

28 
38 

50 

58 
65 

20 
29 
39 

52 

60. 
67 

21 
30 

40 

.8 
1.0 
1.5 

66 

77 
97 

53 

77 
145 

71 

81 
98 

57 
81 
148 

74 

83 
99 

59 
83 
149 

76 
85 
100 

61 
85 
150 

78 
86 
100 

62 
86 
151 

2.0 
3.0 

112 
132 

223 
394 

111 
126 

221 
379 

110 
123 

220 
370 

110 
122 

219 
365 

110 
121 

219 
362 

n 

=  0.020 

1  in 

20000 

1  in 

10000 

1  in 

5000 

1  in 

2500 

1  in 

1000 

Vr 

c 

cVr 

c 

cV  r 

c 

cV  r 

C 

cVr 

c  c 

Vr 

A 
.6 
.8 

32 
44 
55 

13 

26 
44 

36 

48 

58 

14 

29 
47 

39 

51 
61 

15 
31 

49 

41 
53 
63 

16 
32 
50 

42 

54 
64 

17 

33 
51 

1.0 
1.5 
2.0 

64 

82 
95 

64 
123 

190 

67 
83 
94 

67 
125 

189 

69 

84 
94 

69 
126 

188 

71 
84 
94 

71 

127 

187 

72 

85 
93 

72 
127 
187 

3.0 

4.0 

114 
127 

342 
506 

109 
119 

328 

475 

107 
114 

320 
458 

105 
112 

315 

448 

104 
110 

312 
442 

n 

=  0.025 

1  in 

20000 

1  in 

10000 

1  in 

5000 

1  in 

2500 

1  in 

1000 

Vr 

c 

cVr 

c 

cVr 

c 

c  V  r 

c 

cVr 

c  cV  r 

A 
.6 
1.0 

24 
34 
49 

10 
20 
49 

26 
36 
52 

11 

52 

29 
39 
54 

11 
23 
54 

30 
40 
55 

12 
24 
55 

31 
41 
55 

12 
25 
55 

1.5 
2.0 
3.0 

65 
77 
94 

97 
153 

281 

66 
76 
90 

99 
152 

270 

67 

76 

87 

100 
151 
263 

67 
75 
86 

100 
151 
259 

67 
75 
85 

101 
150 
256 

4.0 

106 

423 

99 

396 

95 

380 

93 

371 

91 

366 

n 

-  0.035 

1  in 

20000 

1  in 

10000 

1  in 

5000 

1  in  2500 

1  in 

1000 

Vr 

c 

*Vr 

c 

cV  r 

c 

cVr 

c 

cV  r 

c  cV  r. 

.5 
1.0 
2.0 

19 
34 
55 

9 
34 
110 

21 
35 
55 

10 
35 

109 

''2 
36 
54 

11 
36 

109 

23 
37 
54 

11 
37 

108 

38 
54 

12 

108 

3.0 
4.0 

70 
81 

209 
322 

67 
75 

201 

302 

65 

72 

195 

289 

64 
70 

192 

282 

63 
69 

190 

277 

INDEX 


Aluminum — 
cables,   156 
wire.   155 

Ammonia  vapor,   150 

Angles — 

conversion   of,    78 
functions   of,   31,40 
steel,   116 

Angular  velocities,  84 

Arcs,   circular,   45,   46 

Areas — 

conversion   of,    78 
of  circles,   49,   51,   52 
of  segments,   45,   47 
of   sections,    102 

Atomic  weights,   134 


Beams — 

mechanics  of,   129 

table,   131 

T,  124 

I,    110 

bulb,   112 

Bending  moment,  129,  131 
Boiling  points,   139 
Boyle's  law.   140 
Bulb  beams,   112 


Cables,   156 
Capacity — 

carrying,    154,    155,    156 

specific  inductive.  153 

of   transmission    lines,    158 

of  wires,   154,   155 

of  cables.   156 

units  of.   76 
Catenary,    99 
Center — 

of  mass,   102,   107 

of  pressure,   161 

Centigrade    to    Fahrenheit,     136 
Channels — 

flow  in.  165 

steel,  114 

Charging  current,   158 
Charles'  law,  140 
Circles,   49,   51,   52 
Circular- 
arcs.    45.    46 

cords,    45,    47 

segments.   45,   47 
Circumferences  of  circles,  49,  52 
Coefficients— 

of  contraction,  167 

for   curves,    174 

of  deflection,  109 

of  efflux,   166 

of  elasticity,   128 

for    elbows,    175 

of  expansion,  138 

of  flow,   162 

Kutter's,    174 

for   orifices.    166.   167 

for   pipes,    171.    173 

of  strength,   109 

temperature.    152 

for  weirs,  168,  169,   170 


Common  logarithms,   3.   10 
Columns,  128 

Compound  rolled  shapes,  109 
Compression,    128 
Contraction — partial,    162 
Conversion    factors,    78 
Copper — 

cables,   156 

wire,    154 

Cords  of  circles,   45,   47 
Cosine — 

hyperbolic,   89 

trigonometric,    31 
Cotangent — trigonometric,    31 
Cotton-covered  wire,   57 
Crane — traveling,   133 
Cube  roots — 

of  numbers,  52 

of  fractions,    72 
Cubes  of  numbers,  52 
Cubic  equation,  91,  92 
Curves — 

plane,    99 

flow   through,    165,    17-4 
Cycloid,   99 


Decimals — 

of  an  inch,   73 

of  a  foot,   73 
Definite  integrals,  98 
Deflection  of  beams,  109,  131 
Density — 

of  gases,  141 

of  substances,   134 

of  water,   139 

units  of,   81 

Deposition — electric,    153 
Descartes'   rule.   92 
Discharges — water,    80 
Ditches— flow   in,    165 


Elbows.   165,   174 
Efflux,  162,  166 
Electrolytic   deposition,    153 
Electrostatic    to    electromagnet- 
ic.  84 

Elements — chemical,  134 
Equations — 

of  curves,   99 

solution    of,    90 
Error — probable,   100 
Evaporation — factors  of,   142 
Expansion — coefficients  of.  138 


Factors  of  evaporation.   141.   142 
Fahrenheit  to  Centigrade.   136 
Fifth  roots  and  powers,   72 
Flow — 

coefficients  of,  162 

in   channels.   165 

in  elbows,  165,  174 

through   orMfices,    162 

in  pipes,   164 

over  weirs,   163 
Flumes — flow  in,   165 
Forces — units,   81 


178  INDEX 


Fractions — 

of    an    inch,    73 
of  a   foot,    73 
roots  of,  72 
Francis'   formula,    163 
Fusing  of  wires,   153 


Gas  constants,  141 
Gases,  140 
Gauges — 

B.   &  S.,  157 

wire  and  sheet  metal,  85 
Greek  letters,   2 
Guldin's  rules,   101 
Gyration — radius  of,  102,  109 


Heat   units,   82 
Horner's  method,  92 
Hyperbolic  functions,   87 
Hysteresis,   151 


I-beams,   110 

Immersed  rectangle,   161 

Impedance,   158 

Imperfect    contraction,    162,    167 

Inches — decimals  of,  73 

Inertia — moments  of,   102,  109 

Integrals,   96,   98 


Jet — impact  of,  161 


Kutter's — 

coefficients,   174 
formula,   165 


Moments  of  inertia,  102,  107,  109 
Multiples  of  if  ,   e   and  g,   86 


Natural — 

logarithms,    3,    6,    8 
sine,   cosine,   etc.,   31 

Newton's  method,  94 


Oblique  triangles,  43 
Orifices,  162,  166,   167 


Partial  contraction,   162 

Pipes,    164.    171,    172,    173 

Plane   triangles,   43 

Power — units,     83 

Powers  of  T  ,   e  and  g,   86 

Pressure — 

center  of,  161 
units  of.   82 

Prismoidal  formula,  101 

Probable  error,  100 

Properties   of — 

rolled  angles,  116 
rolled    channels,    114 
I-beams,    110 
rolled  shapes,   109 
T-beams,  124 
bulb  beams,   112 
Z-bars,    126 
copper  cables,   156 
copper  wires,   154 
aluminum   cables,    156 
aluminum   wires,    155 


Latent  heat,  140 
Law — • 

Boyle's,    140 

Charles',   140 

of  B.  &  S.  gauge,  157 
Length — 

of  arcs,  45,  46 

of  cords,   45,   47 

units  of,   78 
Linear   velocities,    84 
Logarithmic    spiral,    99 
Logarithms — - 

common,   3,   10 

natural,  3,  6,  8 
Loss — 

in  curves,  165 

in   elbows.    165 

of  head,   164 

by  hysteresis,  151 


Magnetic  properties,   151 

Materials- — 

strength  of,  128 
density   of,    134 

Melting  points,    139 

Metric   units,   75 

Minutes   to   degrees,   40 


Radius  of  gyration,   102,  109 
Rate    of    electrolytic    deposition, 

Reactance,    158 

Reactions  of  beams,   129,  131 

Reciprocals  of  numbers,   52 

Resistance — 

of  aluminum  cables,  156 
of  aluminum  wires,   155 
of  copper  cables,  156 
of  copper  wires,  154 
specific,   152 

Rise  of  segments,  45,   47 


Saturated  vapor — 
ammonia.    150 
steam.   143 
sulphur  dioxide.  150 

Section   modulus,    102,    109 

Segments.    45,   47 

Series,    95 

Shear,   128.   129 

Sheet    metal   gauge,    85 

Sine- 
hyperbolic.   88 
trigonometric.    31 

Single   phase   lines,    160 


INDEX 


170 


Slide  rule- 
solution   of  equations   by,   90 
on  wire  table,   157 

Smith's  formula,   163 

Specific- 
heats,  138,  141 
inductive  capacity,  153 
resistances,   152 

Spherical   triangles,   44 

Square  roots —  r 

of  fractions,    72 
of  numbers,  52 

Squares  of  numbers,  52 

Steam  tables,  143 

Sulphur  dioxide — vapor  of,  150 


T-beams,  124 

Tangent- 
hyperbolic,  87 
trigonometric,    31 

Temperature — 

coefficients,   152 
units  of,  84 

Tension,    128 

Three  phase  lines,  158 

Transmission  lines,   158 

Triangles — 

plane,   42,   43 
spherical,  44 

Trigonometric  formulae,   41 

Units— 

U.  S.  and  British,  74 
Metric,   75 
conversion   of,    76,    78 


Valency,    134 
Vapor — 

of  ammonia.   150 

of   steam,    143 

of  sulphur  dioxide,  150 
Velocities- 
angular,   84 

linear,   84 
Volumes — 

of  solids,   107 

of   water,    81 

units  of,  74,  75,  76,  79 


Water- 
boiling  point  of,   139 
density   of,    139 
discharges,   80 
flow  of,    162 
pressure   of,   161 
weights  and  volumes,  81 

Weights- 
Metric,  75 
of  materials,  134 
U.   S.   and  British,   74 
units  of,  74,  75,  76,  80 

Weirs,   163,  168,  169,  170 

Wire- 
aluminum,   155 
copper,   154 
cotton-covered,    157 
fusing  of,   153 
gauges,   85 

Work— units  of,  82 

Z-bars,   126 


180  NOTLS 


NOTES  181 


182  NOTES 


NOTES  183 


184  NOTES 


NOTES  185 


186  NOTES 


NOTES  1*7 


NOTES 


NOTES  189 


190  NOTES 


NOTFS  191 


192  NOTES 


NOTP:S  193 


194  NOTES 


NOTES  195 


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